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Philosophy 532 and Philosophy 536 were the two seminars I
presented while on leave at the Princeton University
Philosophy Department, Fall, 2002.
Harvey M. Friedman
PHILOSOPHY 532
PHILOSOPHICAL PROBLEMS IN LOGIC
LECTURE 1
9/25/02
"Mathematics is the only major subject which has been given
a philosophically credible foundation."
This is widely accepted, inside and outside philosophy, but
one can spend an entire career clarifying, justifying, and
amplifying on this statement. Certainly a graduate student
career.
This could entail clarifying what is meant by
"mathematics".
Or "major subject". Or for that matter, "subject". Also
"given". And of course "philosophically credible
foundation", or even just "foundation".
We are almost down to the level of something manageable. In
what sense do we have a "foundation" for mathematics? A
"foundation" in the sense that we don't have for other
subjects such as physics, statistics, biology, law,
history, literary/music criticism or philosophy?
Before I get to the principal features of foundations of
mathematics, let us first touch on "what is mathematics?"
Of course, we do not expect any kind of remotely definitive
answer to such a question. But it is common to focus on
just one aspect of this question. How does mathematics
compare to the (other?) sciences in terms of its
relationship to "reality"?
Let me focus this a bit. Nobody(?) would compare physics
with, say, the study of "the causes of the U.S. Civil War".
In one case, we are endeavoring to uncover "fundamental"
facts about "external reality", which can be "tested" by
repeated "objective" experiments. In the other case, we are
doing something radically(?) different. 2
We start with the most extreme(?) view.
1. Mathematics is just like any other science in that it is
about objects of "nature", that are not man made, but part
of external reality, like physics or other physical
sciences. As a consequence, every properly formed assertion
about the "objects" of mathematics is true or false in the
same sense that a properly formed assertion about physical
reality is true or false. This is usually referred to as
Platonism. This view was championed by Kurt Goedel.
Look at his "What is Cantor's continuum hypothesis?" and
"Russell's mathematical logic".
Most people are uncomfortable with 1. For example, there
remain specific mathematical questions of a rather
fundamental character, around 100 years old, which continue
to resist solution  and the nature of this resistance
makes one feel uncomfortable with Platonism. Specifically,
the continuum hypothesis:
CH: every set of real numbers can be mapped oneone into a
set of natural numbers or onto the set of all real numbers.
Mathematicians have long since dodged this question by
saying things like
"we don't care about every set of real numbers  just the
relatively explicit sets of real numbers that come up in
actual mathematics (the mathematics that we like) like
Borel set of real numbers"
"if we restrict to, say, Borel sets of real numbers, and
Borel maps, then CH is provable"
"CH is abnormal in its use of arbitrary sets of real
numbers, and if you stay within reasonably explicit sets of
real numbers, then you will be safe from logical
difficulties, and you won't have to do any philosophical
thinking"
But it is not reasonable(?) for a philosopher holding
Platonism to dodge CH in this way.
What is most disconcerting is that it has been established
that CH cannot be settled on the basis of the currently 3
accepted axioms for mathematics (Kurt Goedel and Paul
Cohen). In addition, the most favored new axioms for
mathematics  the so called large cardinal axioms  are
also known to be insufficient for settling CH.
It gets worse. The experimental method plays a completely
crucial a primary role in the physical sciences, and
reinforces Platonism in the context of the physical
sciences. (However, there is now talk of some new physical
theories that may in principle be impossible to verify
experimentally! That will probably be very hard for
mainstream physicists to swallow.)
But what takes on the role of "experimentation" or
"experimental verification" or "experimental confirmation"
in mathematics?
It is now clear with the use of computers, that there is a
viable sense of "experimental verification", and more
commonly, "experimental confirmation" in play now in
mathematics.
However, this has appeared only in the context of very
concrete mathematical problems. At present we have no idea
how we can "experimentally verify" or even "experimentally
confirm" anything even remotely like CH through the results
of computation.
There have been attempts to use a much more liberal notion
of "experimental confirmation". Maybe the CH allows us to
solve other mathematical problems, or some group of
mathematical problems, in a direction which somehow strikes
us as much more plausibly true than false. This is
certainly far weaker than what we have in the physical
sciences, where results of experimentation are supposed to
be unarguable(?). Despite some continuing efforts by some
along these lines, in favor of not(CH), the overwhelming
majority of mathematical logicians have not found this
persuasive.
For these reasons (and undoubtedly others), most
mathematical logicians feel much more comfortable with a
hybrid view:
2. Views to the effect that Platonism is correct but only
for certain relatively "concrete" mathematical "objects". 4
Other mathematical "objects" are man made, and are not part
of an external reality.
This is a hybrid point of view and is probably the most
common point of view among mathematical logicians  to the
extent that they are concerned with philosophical issues.
Examples are "finitism" (William W. Tait) and
"predicativism" (Solomon Feferman) and "ultrafinitism"
(Edward Nelson).
(Assuming time permits today, I will come back to some of
the principal isms. This is a substantial topic, and one
can easily arrange an entire seminar on such isms).
There are many possible "isms", some of which have been put
forth, and some of which appear to be implicit in the
"views" of people who do not consider the explication of
"views" as part of their professional activities.
We now come to the main tool that Platonists use for
defense (attacking the opposition). They ask
"where do you draw the line?"
The point is that wherever you draw the line, there is a
natural slightly higher place, and one has to defend why
the stuff on one side of the line is OK whereas the stuff
just barely on the other side of the line is not OK.
Let me give an example. I have seen some ultrafinitists go
so far as to challenge the existence of 2100 as a natural
number, in the sense of there being a series of "points" of
that length. There is the obvious "draw the line"
objection, asking where in
21,22,23,...,2100
do we stop having "Platonistic reality"? Here this ... is
totally innocent, in that it can be easily be replaced by
100 items (names) separated by commas.
I raised just this objection with the (extreme)
ultrafinitist Yessenin Volpin during a lecture of his. He
asked me to be more specific. I then proceeded to start
with 21 and asked him whether this is "real" or something to
that effect. He virtually immediately said yes. Then I
asked about 22, and he again said yes, but with a 5
perceptible delay. Then 23, and yes, but with more delay.
This continued for a couple of more times, till it was
obvious how he was handling this objection. Sure, he was
prepared to always answer yes, but he was going to take 2100
times as long to answer yes to 2100 then he would to
answering 21. There is no way that I could get very far with
this.
To recapitulate, this was used as a defense against the
defense of Platonists that asks for a place to draw the
line between reality and convention(?)  at least in this
one context or related contexts involving large finite
objects. However, I have never seen this kind of defense
used in infinitary contexts in connection with Platonism,
but it may be applicable.
Incidentally, there is a formal line of investigation that
is very relevant here, with some somewhat surprising
outcomes.
I'll first give a formulation using a very weak set theory,
T. In this set theory, all objects are sets, and we have
membership and equality. There are only three axioms:
1. Two sets are equal if and only if they have the same
elements.
2. There is a set with no elements.
3. If x,y are sets then there is a set whose elements are
exactly the elements of x together with y itself.
How big a set can you prove exists in T? Obviously any
size. E.g., each of Ø,{Ø},{{Ø}},... as long as you want.
However, this misses the point. Consider the one with 2100
pairs of braces. One cannot even give a humanly digestible
description of this set in the language of T, let alone
give a humanly digestible proof of its existence in T. I
may be wrong here, but both claims seem likely to be true
even if abbreviations are allowed. (I will see if I can
prove this).
What we really want to know is if there is a "short"
description of a set in T, with a "short" proof in T that
it exists, but where the set is extraordinarily large in 6
some way. We will simply ask that it be extraordinarily
large in the usual sense.
The answer is yes. The result works for T, but the
estimates involved are much nicer if we allow abbreviations
in the underlying logic. Use of abbreviations is completely
standard in virtually any kind of mathematics. We can make
the following definition for 0 ≤ i ≤ 8, in T, in an
especially nice way if abbreviations are allowed. A set has
"rank at most 0" if and only if it is empty. A set has
"rank at most 1" if and only if all of its elements have
"rank at most 0". Continue to: a set has "rank at most 8"
if and only if all of its elements have "rank at most 7".
In fact, there is a much more direct way to define "rank at
most 8" in T, even without any abbreviations. x has "rank
at most 8" iff
there is no y1 Œy2 Œ y3 Œ y4 Œ y5 Œ y6 Œ y7 Œ y8 Œ y9
y1,y2,y3,y4,y5,y6,y7,y8,y9 Œ x. with One can write out a proof in T, especially with
abbreviations, in a mathematically friendly way, with
friendly length and structure. Yet the number of elements
in the set of sets of rank at most 8 is 2^2^2^2^65536.
In the case of arithmetic, one can prove related results,
although I don't know how to prove a result as satisfying
as the above. It is clear how to prove such a result if,
say, addition and (base 2) exponentiation are primitives.
However, in the above case of finite set theory, no
additional primitive was needed to very simply define the
huge set in question.
Can we prove such a result in arithmetic with only 0,1, and
addition? The answer is no. The sizes do grow quite large,
but not the kind of iterated exponentiality that we see
above.
However, it can be done with 0,1,+,x. The example of the
huge number is perhaps not as fundamental as the huge set
above, but reasonably natural. Use the relation
"every 1 ≤ i ≤ n divides m"
to bootstrap up starting at 3. To define the relevant 9
integers, we have to take the least m at each stage. 7
However, it is not obvious that we can prove the existence
of such a least m with an appropriately short (or even
friendly) proof. However, that becomes part of the result.
It would be interesting to do a thorough proof theoretic
investigation of such systems. The only way to get such
huge objects out of such small proofs and such weak axioms
is to be using the cut rule over and over again.
Whoops! I don't know if you can reduce this to, e.g., a
single cut, and at what cost. That is the kind of thing
that I mean by a proof theoretic investigation.
Let us return now to "what is mathematics". We have had a
brief discussion concerning Platonism and Platonism in its
restricted forms.
But even if we settle on such ontological issues, there
still remains a question of an entirely(?) different sort.
It is clear that mathematicians do not even conceive of the
idea of settling the truth value of all sentences about the
mathematical objects that they are concerned with. They are
only interested in certain sentences on a case by case
basis.
Obviously, they cannot be expected to be interested in any
specific sentences of inhuman size. However, sentences of
inhuman size can be interesting to them if they are somehow
incorporated in a sentence of human size.
For example, some mathematicians are quite interested in
arbitrarily (first order) sentences about the field of real
numbers. There are infinitely many of these, and therefore
ones beyond any given finite size, yet all of them come
under a celebrated theorem of Tarski with his decision
procedure for determining the truth value of all of them.
Here arbitrary such sentences get folded into a single
overarching theorem.
The sociology of this is more complicated. Since the
predicate calculus is too philosophical for comfort, the
mathematicians know of Tarski's result in different
notation.
So first of all, the mathematicians can only be (directly)
interested in mathematical statements of quite small size, 8
with abbreviations allowed. Of course, the number of such
statements is absolutely enormous.
Secondly, of even the statements of quite small size (with
abbreviations), only a tiny fraction are "mathematically
interesting".
So any reasonably complete account of what mathematics is,
or what mathematical activity is, must ultimately confront
the issue of what mathematicians are trying to accomplish at least if it is to be relevant to actual mathematical
activity.
This is very difficult to get a hold of, especially in
light of the fact that mathematicians are not in anything
like full agreement as to what they are trying to
accomplish.
What makes matters more difficult still, is that writing
about "what is mathematics, and what are we trying to
accomplish" is not considered normal professional activity
among mathematicians. This statement is not so negative.
After all, "what is philosophy, and what are we trying to
accomplish" is rarely a topic of the leading philosopher's
papers either. I gather that philosophers did not like it
when Rorty wrote about his. I asked Kripke if he would
write about this and related matters, and it he made it
clear that he wouldn't touch it with a xxxxxxxx foot pole!
You do sometimes see such writings by mathematicians, but
about fragments of mathematics, and aimed at very limited
audiences, with very specific purposes in mind. For many
reasons, including the way that fundamental issues are (or
are not) handled, this proves not to be generally useful
for philosophers trying to uncover "what is mathematics,
and what are mathematicians trying to accomplish."
I do get the feeling that this situation is better in
physics. This may be partly due to the apparent fact(?)
that it is relatively clear from the outset "what physics
is and what physicists are trying to accomplish" than "what
mathematics is and what mathematicians are trying to
accomplish".
Nevertheless I think that it is possible to find some
overarching powerful motivating principles for mathematics,
or at least several "brands" of mathematics that would be 9
illuminating, especially to philosophers. Doing this well
for even, say, mathematics before 1950, or even mathematics
before 1900, would be a major achievement.
Coming back to more traditional philosophical concerns,
let's go back to the hybrid view:
2. Views to the effect that Platonism is correct but only
for certain relatively "concrete" mathematical "objects".
Other mathematical "objects" are man made, and are not part
of an external reality.
Under such a view, what is to be made of the part of
mathematics that lies outside the scope of Platonism?
An obvious response is to reject it as utterly meaningless.
Let us then call this an exclusionary view. (We will
discuss more moderate views later). We have already
discussed the "where do you draw the line" attack against
the exclusionary view.
But another problem comes up for the exclusionary view.
What if it turns out that the excluded objects can be used
to derive information about the blessed objects that cannot
be derived just using the blessed objects?
Strictly speaking, we are confusing epistemology with
ontology, and so let me rephrase this.
What if it turns out that accepted axioms associated with
the excluded objects can be used to derive information
about the blessed objects that cannot be derived just from
the accepted axioms associated with the blessed objects?
There is just such a result of Kurt Goedel that applies in
most such situations. You can call it "Goedel's principle".
GP: you can prove the consistency of accepted axioms about
certain objects using accepted axioms about slightly more
"potent" objects, but not with just accepted axioms about
the original objects.
Here "potency" normally coincides with size, or cardinality
in the sense of Cantor. 10
Incidentally, there are more subtle kinds of "potency" that
are very relevant.
We have also seen size considerations play an important
role in the finite. However, a full development of GP in
this context needs to be done. I got involved in what I
call finite Goedel's theorem, which is a decent start.
You can easily argue that GP is not a strong argument
against the exclusionary view. Granted, something is to be
gained by admitting more potent objects, but the gains may
be ill gotten.
Perhaps the "accepted" axioms for the excluded objects are
inconsistent (even short inconsistencies). Then they can
prove anything. Incidentally, attempts to segregate
inconsistent parts of theories from consistent parts of
theories in an appropriate way have not been convincing).
NOTE: There may be a reasonably uniform way of passing from
objects to "standard axioms about them" emerging out of
current research. The idea is to consider all short
sentences about them in primitive notation which are true.
These may be susceptible to complete classification, and
coincide in many cases with the axioms that are intuitively
compelling. There may be more subtle considerations than
"shortness". This is beginning to be carried out with set
theory in primitive notation, and also in arithmetic. END
But the exclusionary view becomes really painful to hold if
it turns out that the use of the excluded objects turns out
to be so effective that lots of mathematics of the kind
valued by mathematicians has to be thrown in the trash.
In fact, implicit in Goedel is the following conjecture:
GP*. you can prove lots of mathematically "interesting" or
"beautiful" facts about certain objects using accepted
axioms about more "potent" objects, but not with just
accepted axioms about the original objects.
In foundations of mathematics, we are at a place where we
are just beginning to assemble a variety of results
supporting GP*. It is still very early, so it is not yet
clear just how pervasive this phenomena is. Obviously this
requires substantial feedback from the mathematical
community for confirmation. 11
I gave a four lecture series called "Rademacher Lectures"
last week at the U Penn Math Dept, on just this topic.
That's why we didn't meet the first week. These lectures
will be on the web shortly.
Look at the following interesting article:
Hilary Putnam, "Philosophy of mathematics: why nothing
works." In: Words and Life, ed. Conant.
PHILOSOPHY 532
PHILOSOPHICAL PROBLEMS IN LOGIC
LECTURE 2
10/2/02
We now discuss some special features of the foundations of
mathematics which illustrate its distinctive power as
compared to attempts at the foundations of other major
subjects. I will make a number of numbered points about
f.o.m.
1. In f.o.m. we have a very simple ontology that suffices
to express all mathematical concepts in a uniform manner.
This simple ontology is that of set theory, with sets,
membership between sets, and equality between sets.
2. There is usually an intuition behind most mathematical
concepts that is not preserved when expressing them in
terms of sets. For instance, nobody declares that natural
numbers are really sets, or that real numbers are really
sets.
3. However, no one seems to know what a natural number
really is, or a real number really is, least of all
mathematicians (smile). The Russellian idea that a natural
number really is a special kind of class of sets is
certainly not compelling either.
4. So under these circumstances, it is clear that the
relationship between mathematical entities is what counts,
and nothing else.
5. Attempts to found mathematics through relationships
only, and not on actual objects, has been a persistent but
elusive goal. Sooner or later, all attempts at such an
autonomous foundation have failed, usually on these 12
grounds: At some point, objects of some sort have to be
postulated, and one has to mirror the standard approach
through set theory. This particularly applies to so called
categorical "foundations" (through category theory, which
is undoubtedly at least a useful organizational scheme for
a considerable variety of mathematical contexts).
6. Coming back to the standard set theoretic f.o.m., since
only the relationship between mathematical entities is what
counts, we first need to judge standard f.o.m. in these
terms. But before we discuss this, we need to give an
overarching description of the nature of this standard
f.o.m.
7. As stated earlier, in standard f.o.m., every
mathematical object is a set. Two sets are equal if and
only if they have the same elements. Thus the only thing
that counts about a set is what's in it. Since everything
is to be a set, all elements of sets are also sets.
8. The simplest of all sets is the set with no elements,
called the empty set, ∅. We can take the set consisting of
just the empty set, written {∅}. We now have two sets,
∅,{∅}, and so we then have the two sets {{∅}},{∅,{∅}}, for
a total of 4 sets. We can continue in this way, obtaining
the so called hereditarily finite sets.
9. The above is supported by that very weak set theory we
have encountered earlier, which we called T. This has the
axioms of extensionality, emptyset, and the adjoining
axiom, x » {y}.
10. Finitary mathematical objects are identified with
hereditarily finite sets. These include natural numbers,
integers, rational numbers, finite sets of rational
numbers, functions from finite sets of rational numbers
into finite sets of rational numbers, Cartesian products of
finitely many sets of rational numbers, polynomials of
several variables with rational coefficients. But
definitely not real numbers and complex numbers, which are
infinitary in an essential way. With more subtlety, one
also handles certain rather special real and complex
numbers, particularly real algebraic numbers and complex
algebraic numbers.
11. Set theory really gets going when we go further than
the hereditarily finite sets. We can take the set of all 13
hereditarily finite sets. This is supported by the axiom of
infinity, although the usual axiom of infinity is too weak
to get this without some additional axioms.
12. The set of all hereditarily finite sets (usually called
HF) together with its subsets, is more than enough to
handle the most common infinitary objects in mathematics.
These include real and complex numbers, and more generally,
elements from a complete separable metric space. All of the
usual arguments of "countable" mathematics can be easily
done with HF and its subsets.
13. With some care, almost all of the statements of current
mathematics can be appropriately viewed as "countable"
mathematics; the exceptions consist of general
formulations, where the generality of the formulations is
something leading mathematicians stand ready to sacrifice.
14. But how is "having all subsets of HF" reflected in the
axioms? This is by the crucial separation axiom, which says
that the set of all elements of HF, or of any given set,
obeying any given property (parameters allowed) presented
as a formula in the language of set theory.
15. Countable set theory does not directly take care of,
say, the set of all real numbers. Though for almost all
practical purposes, one can just talk about real numbers
without having the set of all such. The set of all real
numbers and more generally, the completion of any countable
metric space, is supported by another f.o.m. axiom, the
axiom of power set. This asserts that the set of all
subsets of any given set exists.
16. We now have a very powerful system, more or less what
is called Zermelo set theory. This supports HF, S(HF),
S(S(HF)), S(S(S(HF))), etc. Also, of course, their subsets.
This is a very powerful set theoretic structure. But it
does not include itself as a set. I.e., we can take the
union of all of these sets just listed, to form a set that
is bigger than anything so far. To support this union, we
need the use the axiom of replacement.
17. There is one more major axiom missing, and that is the
axiom of choice. This was controversial for many years, but
now is considered standard. This says that if you have a
set of nonempty sets, no two of which have any elements in
common, then there is a set that has exactly one element in 14
common with each of them.
18. The full standard f.o.m. is called ZFC for Zermelo
Frankel set theory with the axiom of choice. The axioms
are:
1.
2.
3.
4.
5.
6.
7.
8.
9. Extensionality.
Pairing.
Union.
Separation.
Power set.
Choice.
Foundation.
Replacement.
Infinity. All of 18 hold in HF. The addition of 9 is incredibly
powerful.
19. How adequate is ZFC? It is the de facto standard for
correctness, and will be used if a dispute arises in the
mathematics community as to correctness that is not
resolved informally. But I do not believe that it has ever
had to be used for this purpose  at least for many
decades. Disputes get resolved informally and semiformally
well before ZFC needs to seriously enter the picture.
20. For nearly all mathematical purposes, ZFC is known to
suffice. In fact, for nearly all mathematical purposes,
very weak fragments of ZFC are known to suffice.
21. Detailed studies of principal fragments of ZFC
(including ZFC itself) and what math can be done there and
what math cannot be done there is perhaps the main line of
investigation in modern f.o.m.
22. There are a variety of problems that we now know cannot
be settled in ZFC. However, at least until very recently,
there has been something "abnormal" about these examples.
The abnormalities are sufficiently pronounced that
mathematicians can somewhat defensibly segregate them from
the kind of mathematics that they have most traditionally
valued.
23. In particular, at least until recently, the examples
have been visibly less "concrete" than "normal" 15
mathematics. We are now just beginning to see examples that
are regarded as having a normal level of concreteness, and
judged to be "beautiful" by some of the leading
mathematicians in the world. "Beauty" is a special buzz
word among mathematicians, which is more than enough to get
the so described statement admitted to the realm of "real
mathematics".
***********************
We now discuss Russell's paradox for sets and its usual
ways out. This results in some important formal systems. We
then discuss Russell's paradox for a number of concepts,
and consider analogs of the usual ways out for these
concepts.
Let us begin with the following formal system FCS (full
comprehension for sets). The language is the binary
relation symbol Œ only. The nonlogical axioms consist of
($x)("y)(y Œ x ´ A)
where x is not free in the formula A.
This system FCS is inconsistent, as can be seen by
replacing A with y œ y, thereby obtaining
($x)("y)(y Œ x ´ y œ y).
Let x be such that
("y)(y Œ x ´ y œ y).
Then
xŒx´xœx
which is a contradiction.
The refutation of the once seemingly innocent FCS is called
Russell's paradox for sets.
One of the most natural ways out involves using two sorts
of variables. Lower case letters for objects, and upper
case letters for sets. The atomic formulas are of the form
x Œ A, where x is an object variable and A is a set
variable. The appropriate comprehension axiom now reads 16
($A)("x)(x Œ A ´ j)
where j is any formula in which A is not free.
Of course, this is far too weak by itself to be a good way
out since one cannot even prove that there exists more than
one object.
Actually that last sentence makes no sense, since we don't
have equality. So how do we formulate it? We cannot even
prove that there exists two sets, with an object in one
that is not in the other.
For any usual purpose for f.o.m., we would like some sort
of axiom of infinity. However, there is no way to express
anything like the axiom of infinity in this language. More
precisely, every consistent extension of this two sorted
comprehension axiom scheme has a finite model. This follows
from work done on monadic second order theories (Buchi et
al).
The road to appropriately powerful set theories of this
kind is to introduce a third sort. Thus we will have
objects, sets of objects, and sets of sets of objects. We
will use superscripts 0,1,2, respectively. The atomic
formulas are of the form x Œ y, where the superscript on x
is one less than the superscript on y.
The comprehension axiom scheme we are interested in reads
($xi+1)("yi)(yi Œ xi+1 ´ j)
where j is a formula in the three sorted language in which
xi+1 is not free. Here i = 0 or 1.
Of course this is still too weak, since it has a finite
model.
But now we can formulate a reasonable form of the axiom of
infinity. Informally, this asserts that there is a nonempty
set of sets of objects in which every element is properly
included in an element. We leave it as an exercise to
formalize this in our three sorted language.
This gives us a reasonably powerful system which we will
call IST(3) + INF, or 17
impredicative set theory with 3 types
plus the axiom of infinity.
QUESTION: Is this form of the axiom of infinity the
"simplest" possible axiom consistent with our comprehension
axiom scheme, that forces all models to be infinite?
IST(3) + INF is obviously interpretable with the system Z3
of third order arithmetic that we will encounter in PHIL
536 in a few weeks. Z3 also has three sorts, and the
intended interpretation is that the objects of the first
sort are exactly the natural numbers; of the second sort,
the sets of natural numbers; of the third sort, the sets of
sets of natural numbers. We also have 0,S,+,• on the first
sort, with the usual quantifier free axioms, and with
induction for all formulas in the language.
It may seem reasonable that Z3 is interpretable in IST(3) +
INF. However, we are currently checking a proof that this
is false. In fact, we are checking a proof that IST(3) +
INF is mutually interpretable with Z2 and not Z3 (just the
part of Z3 without sets of sets of natural numbers). This is
a bit tricky, and I will discuss this further at a later
date.
However, let us now strengthen the axiom of infinity in the
following way. Let INF* assert that there is a nonempty set
of sets of objects in which every element has a least
proper superset among the elements.
We are also checking a proof that IST(3) + INF* is mutually
interpretable with Z3.
For each n ≥ 2, it is clear how to define the nsorted
system IST(n). The result should be that for n ≥ 3, IST(n) +
INF is mutually interpretable with Zn1, and for n ≥ 3,
IST(n) + INF* is mutually interpretable with Zn.
Let IST(<•) be the union of the IST(n), n ≥ 2. Let Z(<•) be
the union of the Zn, n ≥ 2. Then IST(<•) + INF, IST(<•) +
INF*, Z(<•) are mutually interpretable. This is well known
without my recent efforts.
IST(<•) + INF or INF* is a streamlined version of what is
essentially due to Russell, although he wrote down a far
more complicated formalism, dividing the system into a 18
purely predicative part and an axiom of reducibility that
allows the derivation of impredicative comprehension axiom
schemes.
The formalization of mathematics using such many sorted
systems as IST(<•) + INF* is cumbersome and unpleasant.
This is why f.o.m. turned to axiomatic set theory.
There is a set theory which corresponds very closely to
IST(<•) + INF. This is the so called Zermelo set theory, Z.
The language of Z is Œ,=. The axioms are
1. Extensionality. If two sets have the same elements then
they are equal.
2. Pairing. The set consisting of any two given sets
exists.
3. Union. The set consisting of the elements of the
elements of any given set exists.
4. Separation. The set consisting of all elements of any
given set that satisfy any given condition expressible in
Œ,= exists.
5. Power set. The set consisting of all subsets of any
given set exists.
6. Infinity. There is a set A with Œ A and for all x Œ A,
x » {x} exists and lies in A.
A remark about Separation. We chose to give this
informally, but a slight ambiguity has arisen. The formal
version is
($y)("z)(z Œ y ´ z Œ x Ÿ j)
where y is not free in j and j is a formula in Œ,=.
From the informal description above, it was perhaps not
clear that parameters are allowed in j representing
arbitrary sets. So we are using conditions expressible with
the aid of reference to any sets, regardless of how these
sets (parameters) can be themselves defined. This is
typical of formal systems for f.o.m. purposes, although
parameterless separation and comprehension has come up in
important f.o.m. contexts.
IST(<•) + INF* is interpretable in Z. However, Z is not
interpretable in IST(<•) + INF*, and in fact Z proves the
consistency of IST(<•) + INF*. However, Z is not much 19
stronger than IST(<•) + INF*. To indicate this, let us
consider a weakening of Z called BZ (bounded Zermelo).
We call a formula of set theory (Œ,=) bounded if and only
if all of its quantifiers are bounded in the following
sense.
A bounded quantifier is usually written in one of two ways:
("x Œ y)
($x Œ y).
These are of course meant to be abbreviations. I.e.,
("x Œ y)(j) abbreviates ("x)(x Œ y Æ j)
($x Œ y)(j) abbreviates ($x)(x Œ y Ÿ j).
In BZ, we use exactly the same axioms, except that
Separation is weakened to require that the formula used be
bounded.
BZ, it turns out, is finitely axiomatizable, as opposed to
Z or IST(<•) + INF (INF*).
IST(<•) + INF* is interpretable in BZ, but BZ is not
interpretable in IST(<•) + INF*. The later is clear because
otherwise, BZ would be interpretable in IST(n) + INF* for
some n, which contradicts that BZ proves the consistency of
IST(n) + INF*.
Here are the results I
+ INF*, IST(<•) + INF*
fragment of arithmetic
consistencies of these
equivalent. am getting at. Firstly, BZ, IST(<•)
are equiconsistent in that a weak
suffices to prove that the
three individual systems are Secondly, BZ and BZ, (IST(<•) + INF*)', (IST(<•) + INF*)'
are mutually interpretable, where ' is as discussed in PHIL
536.
Thirdly, there is a natural interpretation of Z<• (the union
of the Zn) into each of these three systems. The result is
that if j is a sentence in the language of Z<•, and T is any
of the three systems, then j is provable in Z<• iff the
interpretation of j is provable in T.
This result certainly does not hold of Z. 20
The extension of Z to the current gold standard foundation
for mathematics, ZFC, arose out of the glaring
incompleteness of Z.
Look at the series w, S(w), S(S(w)), etc., where S is the
power set operation, and w is the least set obeying the
condition in the axiom of infinity. One cannot obtain the
union of these sets in Z. To remedy this, the axiom of
Replacement was introduced:
7. Replacement. Suppose that an operation assigning a
unique set to any element of a given set is expressed in
Œ,=. Then the set of all sets so assigned exists.
Another glaring incompleteness is more conceptual. We will
come to this after we have discussed the so called
cumulative hierachy of sets.
The idea behind IST(<•) is that the sets are arranged in an
organized linearly ordered hierarchy, where every set
consists entirely of lower sets.
This conceptual picture is easy to maintain in connection
with Z. In fact, one gets a cumulative hierarchy over w,
indexed by the natural numbers. First the elements of w,
then the sets of those, then the sets of those, etc. Since
w is a transitive set (every element of an element is an
element), this cumulates. I.e., every set appearing at any
given stage appears at all subsequent stages.
For conceptual homogeneity, it has been standard to not use
w as the starting point, but rather ∅. For infinitely many
stages, one has only finitely many sets at each stage.
I.e., we are looking at
∅, S(∅), S(S(∅)), etc.
Note that they grow very fast, and in fact the sizes
successively exponentiate to base 2.
At the end of these first infinitely many stages, one takes
the union of the sets in this hierarchy, the union being
called HF = the set of all hereditarily finite sets.
NOTE: HF does not stand for "Harvey Friedman", as much as I
would like it to (smile). 21
However, Replacement is needed to construct this hierarchy
∅, S(∅), S(S(∅)), etc
and
the
set
x,y prove that HF exists. A good alternative is to sharpen
axiom of infinity to assert the existence of a nonempty
containing ∅, and closed under the operation that sends
to x » {y}. Coming back to the standard cumulative hierarchy, we first
begin gently with V(0) = ∅, V(n+1) = S(Vn), indexed along
the natural numbers. It is usual to define V(w) = HF to be
the union of these V(n). Then we continue with V(w+1) =
S(V(w)), V(w+n+1) = S(V(w+n)), n < w.
The union of the V(w+n) is written V(w+w).
The construction of V(w+w) cannot be done in Z, even if we
were to sharpen the axiom of infinity in the way indicated
above so that we would be able to get V(w) = HF.
V(w+w), or more precisely (V(w+w),Œ), forms what is
generally called the standard model of Z.
The full cumulative hierarchy of sets is defined in terms
of the notion of ordinal in set theory. These are the
epsilon connected transitive sets. The hierarchy is given
by V(0) = ∅, V(a+1) = S(V(a)), V(l) = »a<lV(a). Here 0 = ∅,
a+1 = a » {a}, and l is a limit ordinal.
There is nothing present in the axioms of Z = Zermelo set
theory that bears on the question of whether all sets
appear in the cumulative hierarchy of sets.
The intention behind the axiom of Foundation is that using
it, one can prove that every set lies in (some stage of)
the cumulative hierarchy.
8. Foundation. Every nonempty set has an element no element
of which is in the given nonempty set. I.e., every nonempty
set has an epsilon minimal element.
Thus we now have what is called ZF:
1. Extensionality.
2. Pairing. 22
3.
4.
5.
6.
7.
8. Union.
Separation.
Power set.
Infinity.
Replacement.
Foundation. The final glaring(?) incompleteness is the axiom of choice.
The axiom of choice is very useful for stating mathematical
theorems in convenient full generality. It is known that
any suitably concrete mathematical theorem that can be
proved with the axiom of choice can be proved without the
axiom of choice. Nevertheless, for the purposes of creating
a reasonably "conceptually complete picture" of the
cumulative hierarchy of sets, we definitely need to have
the axiom of choice.
An alternative would be to construct a conceptually
attractive and "reasonably complete" picture in which the
axiom of choice is contradicted. However, we are very far
from being able to do this.
9. Choice. For every set of pariwise disjoint nonempty
sets, there is a set which contains exactly one element
from each.
We have arrived at ZFC, the gold standard:
1.
2.
3.
4.
5.
6.
7.
8.
9. Extensionality.
Pairing.
Union.
Separation.
Power set.
Infinity.
Replacement.
Foundation.
Choice. It seems like ZFC ought to be complete in some tangible
sense. It has stood the test of time, having emerged in
clear form by 1920. It provides a lot of comfort. Not only
are mathematicians generally comfortable with what it says,
and therefore its presumed consistency, they are also
generally comfortable with its role as the de facto gold
standard. They like to keep it in the background as an
ultimate arbiter, but they rarely need to pull it out for
this purpose. I have heard recently, however, that some 23
people have run into trouble using very abstract category
theory which they never bothered to see how to formalize in
ZFC (or minor extensions of ZFC). They used what amounts to
inconsistent theories, and their results had to be
retracted. (Serves them right! (smile)).
It is also true that mathematicians now don't think much
about ZFC, and very few can recall the axioms. This is
partly from its great success, as it lies in the
background. But another aspect of this is that in normal
mathematics, one uses only a very tiny fragment of ZFC.
Transforming this situation has been one of my major goals.
Coming back to the feeling of "conceptual completeness" of
ZFC, there is a result that makes it perhaps all the more
difficult to get at this.
We found a surprisingly simple statement in primitive
notation (Œ,=) which cannot be decided in ZFC.
For those of you who are familiar with class theory, the
statement in class theory is a bit simpler in that it saves
a quantifier.
PROPOSITION. Every transitive proper class has a four
element chain under inclusion. I.e., there are distinct
elements x,y,z,w such that x Õ y Õ z Õ w.
This is independent of even the Morse Kelley theory of
classes with the global axiom of choice (for those of you
who know what I am talking about).
Here is the set theoretic version.
PROPOSITION. The transitive sets that do not have a four
element chain under inclusion form a set.
This is neither provable nor refutable in ZFC. It uses 7
quantifiers (in Œ,=), but there is a way to modify it so as
to use only 5 quantifiers. We also know that every 3
quantifier sentence in Œ,= can be proved or refuted in ZFC.
What about 4? That seems to be a difficult open question.
Nevertheless, I am still optimistic about finding a result
of the following kind for systems like Z, ZC, and ZFC:
"The suitably simple set theoretic statements (including 24
schemes) in primitive notation that are not obviously
inconsistent with the set theoretic framework form an
axiomatization of ZFC." PHILOSOPHY 532
PHILSOOPHCIAL PROBLEMS IN LOGIC
LECTURE 3
10/9/02
We have discussed Russell's paradox for sets, and the
impredicative theory of types as a first way out. This was
replaced for purposes of the foundations of mathematics, by
the closely associated Zermelo set theory (Z). Then Z, in
turn, is then "completed" (in a sense we don't yet
understand) by the gold standard Zermelo Frankel set theory
with the axiom of choice, ZFC.
We now want to discuss several other ways out (of Russell's
paradox for sets), some of which are far more powerful than
ZFC in terms of interpretation power. There is an ongoing
search for new ways out, based on new unifying principles,
which may have compelling philosophical stories that are
superior to the heterogenous and specialized stories that
have been given for ZFC (and the so called large cardinal
axioms).
There is a constant adjustment and modification of these
new ways out according to possible philosophical stories,
as well as interpretation power. One ultimately wants (and
will undoubtedly get) massive interpretation power
sufficient to interpret at least the largest of the large
cardinal axioms studied by set theorists, together with
intriguing and compelling philosophical stories that will
have counterparts throughout analytic philosophy. One
should be able to get a range of interpretation powers,
with a corresponding range of philosophical stories, of
ever increasing boldness.
Here are the ways out that we will discuss.
1. Two sorted theory of classes with sets and classes as
separate sorts.
2. Single sorted set/class theory. 25
3. Small/large set distinction, leading to a range of new
issues.
4. Two set theoretic universes, leading to the ZFC level.
5. Three and more set theoretic universes. 1. Class/set theory.
The standard theory of this kind is NBG, after von Neumann,
Bernays, Gödel.
There are two sorts of variables, lower case over sets, and
upper case over classes. The atomic formulas are of the
form
x=y
xŒy
x Œ A.
Note that we can't even ask the question of whether a given
class is an element of a given set, or even whether two
classes are equal. However, one can, if one wishes, make
perfectly good definitions of these concepts. For the
latter, just that they have the same set elements. For the
former, that there is a set element of the given set that
has the same set elements as the given class.
We first give the axioms of NBG so that they most closely
resemble those of ZF. We then give a well known
simplification of them.
1. Extensionality. If two sets have the same elements then
they are equal.
2. Pairing. The set consisting of any two given sets
exists.
3. Union. The set consisting of the elements of the
elements of any given set exists.
4. Separation. The set consisting of all elements of any
given set that satisfy any given condition expressible in
our two sorted language without class quantifiers, exists.
5. Power set. The set consisting of all subsets of any
given set exists.
6. Infinity. There is a set z with ∅ Œ z and for all x Œ z,
x » {x} exists and lies in z.
7. Replacement. Suppose that an operation assigning a 26
unique set to any element of a given set is expressed in
our two sorted language without class quantifiers. Then the
set of all sets so assigned exists.
8. Foundation. Every nonempty set has an element no element
of which is in the given nonempty set. I.e., every nonempty
set has an epsilon minimal element.
9. Class comprehension (without class quantifiers). The
class of all sets satisfying any condition expressible in
our two sorted language without class quantifiers, exists.
In light of 9, we can simplify the remaining axioms of NBG
as follows.
1. Extensionality.
2. Pairing.
3. Union.
4. Separation. The common elements of any given class and
any given set forms a set.
5. Power set.
6. Infinity.
7. Replacement. The values of any univalent class of
ordered pairs with arguments from a given set forms a set.
8. Foundation.
9. Class comprehension (without class quantifiers).
Note that this formulation has infinitely many axioms
because of the scheme 9. However, it is known how to
replace 9 with finitely instances, thereby giving a finite
axiomatization of NBG. However, this has not been done in
any elegant way.
Here we can utilize one of our main themes. We can show
that we can use all instances of 9 with at most one
quantifier and at most a few set variables and at most a
few class variables. We have not gone into the details of
this. It may be quite difficult to see what minimum sizes
suffice.
We now come to the issue of the axiom of choice. First of
all, there is the ordinary Set Choice, or AxC:
10. Choice (AxC). For any given set of pairwise disjoint
nonempty sets, there is a set which contains exactly one
from each of the nonempty sets.
However, there is the obvious version for classes, called
Class choice, or Global choice (GC): 27
10'. Global choice (GC). For any given class of pairwise
disjoint nonempty sets, there is a class which contains
exactly one from each of the nonempty sets.
GC seems as powerful in class theory as Set choice is in
set theory. Let me amplify on this.
It is universally believed that AxC is the strongest
"choice principle" one can formulate in set theory.
However, nobody knows what this last sentence means! There
should be some general criterion on what a choice principle
is, with a proof that AxC implies any such choice
principle. This has not yet been done.
Evidence of the power of the AxC is that it suffices,
within even Z, to prove the set well ordering principle
every set can be well ordered.
It is also true that GC proves the class well ordering
principle
the class of all sets
can be well ordered.
It would seem that this is the strongest choice principle
in class theory. However this is not true! There is a
choice principle that cannot be derived from GC.
For instance, the choice principle
("A)($B)(j(A,B)) Æ ($A1,A2,...)(each j(An,An+1) holds),
or if one wants to avoid use of the natural numbers,
("x)($A)(j(x,A)) Æ ($B)("x)(j(x,Bx)).
Presumably these choice principles cannot be proved in NBG
+ GC, even if we require that j have no class quantifiers.
The reason that NBG has become so standard, despite its
glaring incompletenesses discussed below, is its
relationship to ZF. Also the relationship between NBG + GC
with ZFC. 28
THEOREM. NBG and ZF prove the same sentences without class
variables. ZF and NBG are equiconsistent. NBG is not
interpretable in ZF. NBG and ZF' are mutually
interpretable.
Recall that NBG is finitely axiomatizable, and ZF is not
finitely axiomatizable. That is what is behind why NBG is
not interpretable in ZF.
Here equiconsistency is taken to mean that
Con(ZF) ´ Con(NBG)
is provable in a weak fragment of Peano arithmetic (PA). In
fact, the fragment associated with superexponentiation
suffices.
Note that ZF' is the result of the general prime operator
on theories that we have discussed in PHIL 536.
THEOREM. NBGC = NB + GC and ZFC prove the same sentences
without class variables. ZFC is a subsystem of NBG + GC. ZF
and NBG are equiconsistent. NBG + GC is not interpretable
in ZFC. In fact, even NBG is not interpretable in ZFC. NBG
+ GC and ZFC' are mutually interpretable.
Except for the first claim, this second theorem follows
from the first theorem since ZF and ZFC are mutually
interpretable and equiconsistent, and NBG and NBG + GC are
mutually interpretable and equiconsistent.
We now come to the glaring incompleteness of NBG. This is
of course the restriction that there be no class
quantifiers in class comprehension.
If we remedy this, we get the system known as MK for the
Morse Kelley theory of classes:
1. Extensionality.
2. Pairing.
3. Union.
4. Separation.
5. Power set.
6. Infinity.
7. Replacement.
8. Foundation.
9'. Class comprehension. The class of all sets satisfying 29
any condition expressible in our two sorted language,
exists.
MKGC is a "reasonably complete" theory of classes, whatever
that means. It is not "as complete" as ZFC, because of the
fact that GC does not derive some choice principles
mentioned above.
2. Single sorted set/class theory. This is simpler than the theory of classes with sets and
classes in that it is single sorted. It is also quite
convenient.
There are two common ways of doing this. The first uses the
language of ordinary set theory with Œ,=,M, where M is a
unary predicate symbol.
M(x) is defined to mean
These are the sets that
theory. The x such that
in class theory that do
set. that x is an element of some set.
correspond to the sets in class
not M(x) correspond to the classes
not have the same elements as any The second way of doing this is essentially the same. We
use Œ,= only. We define M(x) to mean "x is an element of a
set", and repeat the same axioms as we use for the first
way. The idea is that they are to be officially expanded
into primitive notation.
We leave it to you to actually write down the theories
under the two approaches, and to formulate the
relationships between these theories and NBG, MK.
3. Small/large set theory.
Here we have a single sorted theory with Œ,=, and the unary
predicate symbols SM(x) and LA(x), for "x is small" and "x
is large". Every set is either small or large, but not
both.
As opposed to the previous section, we will have large sets
that are elements of sets. In fact, we will have large sets
that are elements of small sets  namely their singleton.
The idea is very obvious. Smallness is a matter of size, 30
and so it is perfectly natural for a large set to be an
element of a small set, or a large set.
We want to give a preferred model(s) for this conception.
To do this, let us back up and give the corresponding
preferred model(s) for ZFC and for MKGC (Morse Kelly with
the global axiom of choice).
Let k be a strongly inaccessible cardinal. This is a von
Neumann cardinal k such that
i) for all a < k, 2a < k;
ii) k > w;
iii) any subset of k of cardinality < k is bounded above.
Then (V(k),Œ) is a model of ZFC. Also (V(k),Œ,V(k+1)) is a
model of MKGC.
The metatheory in which this discussion is carried out can
be taken to be ZFC + "there exists a strongly inaccessible
cardinal", or even ZC + "there exists a strongly
inaccessible cardinal".
The preferred model(s) for this conception is as follows.
The sets consist of all sets that are included in a
transitive set of cardinality £ k. The small sets are such
sets of cardinality < k. The large sets are such sets of
cardinality k.
The natural axioms that hold in this structure are as
follows.
1. Every set is small or large, but not both.
2. Every subset of a small set is small.
3. Extensionality. If two sets have the same elements then
they are equal.
4. Pairing. The set consisting of any two given sets
exists, and is small.
5. Union. The set consisting of the elements of the
elements of any given set exists.
6. Union'. The set consisting of the elements of the small
elements of any given small set exists and is small.
7. Large set. The set consisting of all sets that are a
subset of a small transitive set is a large set.
8. Separation. The set consisting of all elements of any
given set that satisfy any given condition expressible in 31
our language, exists.
9. Power set. The set consisting of all small subsets of
any given set exists.
10. Power set'. The set consisting of all subsets of any
small set exists and is small.
11. Infinity. There is a small set z with ∅ Œ z and for all
x Œ z, x » {x} exists and lies in z.
12. Replacement. Suppose that an operation assigning a
unique set to any element of a given set is expressed in
our language. Then the set of all sets so assigned exists.
13. Replacement'. Suppose that an operation assigning a
unique set to any element of a given small set is expressed
in our language. Then the set of all sets so assigned
exists and is a small set.
14. Foundation. Every nonempty set has an element no
element of which is in the given nonempty set. I.e., every
nonempty set has an epsilon minimal element.
15. Choice. For any given set of pairwise disjoint nonempty
sets, there is a set which contains exactly one from each
of the nonempty sets.
16. Choice'. For any given small set of pairwise disjoint
nonempty sets, there is a small set which contains exactly
one from each of the nonempty sets.
We can interpret MKGC in this system as follows. The sets
in MKGC are the sets that are subsets of small transitive
sets. The classes in MKGC are the sets whose elements are
not subsets of small transitive sets.
We can interpret this system in MKGC as follows. The sets
in this system are identified with the extensional well
founded relations on the class of all sets in MKGC. The
small sets are those whose domain is a set in MKGC.
There are a number of tricky issues concerning sharp
conservative extension results between set and class
theories, versus the above system and its variants.
This axiomatization is an example of where a lot of axioms
are written down about a clearly conceived structure, and
where one can easily omit important axioms by accident.
After intensive study, one generally gets to a finished
product that has some sort of feel of "completeness" about
it.
In every such case, one would like to know just what kind 32
of "completeness" one has, or what kind of "completeness"
one can at least establish. All of the main formal systems
emanating out of f.o.m. should be subject to such an
investigation.
4. Two set theoretic universes.
We now discuss a system with two set theoretic universes,
and two very simple axiom schemes. This system is
equiconsistent with ZFC. ZFC is interpretable in the
following system.
We consider the following system S in L(Œ,W). Here W is a
constant symbol. The system is related to what is called
Ackermann set theory, but it is simpler, and also it lacks
extensionality. We are trying to be minimalistic.
1. SS. x Œ W Æ ($y Œ W)("z)(z Œ y ´ (z Œ x & j)), where j
is a formula in L(Œ,W) in which x is not free.
2. WIT. x1,...,xk Œ W Æ (($y)(j) Æ ($y Œ W)(j)), where k ≥
0 and j is a formula in L(Œ) with at most the free
variables x1,...,xk,y.
SS stands for "subworld separation". We think of W as a
subworld. WIT stands for "witness". I used to write "RED"
for reducibility, in vague connection with Russell's axiom
of reducibility, but I now prefer WIT.
We can derive that W is a transitive set from these axioms.
The significance of this is that every set in the first
world; i.e., W, does not pick up elements from the second,
"outer", world.
We first considered the above system with the axiom
3. EXT. ("z)(z Œ x ´ z Œ y) Æ (x Œ z ´ y Œ z).
We first derived in K(W) all axioms of ZFC except AxC and
foundation, and derived a sharp form of replacement called
reflection. NOTE: in the presence of foundation,
replacement and reflection are equivalent, but not so
without foundation. These derivations are pretty
straightforward and fun! This was enough to conclude that
ZFC is interpretable in SS + WIT + EXT.
We then showed how to interpret SS + WIT + EXT in SS + 33
WIT. This shows that ZFC is interpretable in SS + WIT.
5. Three and more set theoretic universes.
Having three or more set theoretic universes, with an
indistinguishability principle gives some significant large
cardinals (small large cardinals).
We present the system K(W1,W2). The language is L(Œ,W1,W2),
whose nonlogical symbols are Œ,W1,W2, where W1,W2 are
constant symbols.
1. W1 Œ W2.
2. Subworld separation. x Œ W1 Æ ($y Œ W1)("z)(z Œ y ´ (z
Œ x Ÿ j)), where j is a formula in L(Œ,W1,W2) in which y is
not free.
3. Resemblance. x1,...,xk Œ W1 Æ (j ´ j[W1/W2]), where k ≥ 0
and j is a formula in L(Œ,W1) with at most the free
variables x1,...,xk.
4. Extensionality. ("z)(z Œ x ´ z Œ y) Æ (x Œ z ´ y Œ
z).
Obviously we can derive 2 with W1 replaced by W2.
K(W1,W2) is very considerably stronger than ZFC and K(W).
A cardinal k is extremely indescribable iff for all R Õ
V(k) and first order sentence j, if (V(k+k),Œ,R) satisfies
j, then $ a < k such that (V(a+a),Œ,R « V(a)) satisfies j.
THEOREM. ZFC + "there exists an extremely indescribable
cardinal" is interpretable in K(W1,W2).
A cardinal k is subtle iff for all sets Aa Õ a, a < k, and
closed unbounded C Õ k, there
exists a < b from C such that Aa = Ab « a.
THEOREM. K(W1,W2) is interpretable in ZFC + "there exists a
subtle cardinal".
We showed that these results still hold if we cut K(W1,W2)
down considerably as follows.
Let T(W1,W2) be the following system, in the same language
L(Œ,W1,W2).
1. Parameterless comprehension. ($x)(x Œ W2 Ÿ ("y)(y Œ x ´
(y Œ W1 & j))), where j is a formula in L(Œ,W1) with at most 34
the free variable y.
2. Resemblance. x Œ W1 Æ (j ´ j[W1/W2]), where j is a
formula in L(Œ,W1) with at most the free variable x1.
3. Extensionality. ("z)(z Œ x ´ z Œ y) Æ (x Œ z ´ y Œ
z).
The importance of comprehension without parameters is that
it makes sense for what we call pure predication. Thus
instead of thinking of sets, we can instead think of pure
predicates, which do not allow reference to particular
objects. General predication is what most axiom systems are
based on, as well as mathematics itself. Reference is
allowed to any finite list of objects, regardless of
whether these objects can be "defined" in any way.
We never got a chance to see if we can remove
extensionality. It is extremely likely that we can, in a
way that is similar to our removal of extensionality from
K(W).
Going further, we present the system K(W1,W2,W3). The
language is L(Œ,W1,W2,W3), whose nonlogical symbols are
Œ,W1,W2,W3, where W1,W2,W3 are constant symbols.
1. W1 Œ W2 Ÿ W2 Œ W3.
2. Subworld separation. x Œ W1 Æ ($y Œ W1)("z)(z Œ y ´ (z
Œ x Ÿ j)), where j is a formula in L(Œ,W1,W2,W3) in which y
is not free.
3. Resemblance. x1,...,xk Œ W1 Æ (j ´ j[W1,W2/W2,W3]), where
k ≥ 0 and j is a formula in L(Œ,W1,W2) with at most the free
variables x1,...,xk.
4. Extensionality. ("z)(z Œ x ´ z Œ y) Æ (x Œ z ´ y Œ
z).
THEOREM. ZFC + "there exists a subtle cardinal" is
interpretable in K(W1,W2,W3), which is interpretable in ZFC
+ "there exists a 2subtle cardinal".
We can define the systems K(W1,...,Wn), n ≥ 2, as follows.
1. W1 Œ W2 Ÿ ... Ÿ Wn1 Œ Wn.
2. Subworld separation. x Œ W1 Æ ($y Œ W1)("z)(z Œ y ´ (z
Œ x Ÿ j)), where j is a formula in L(Œ,W1,...,Wn) in which y
is not free.
3. Resemblance. x1,...,xk Œ W1 Æ (j ´ j[W1,...,Wn1/W2,...,
Wn]), where k ≥ 0 and j is a formula in L(Œ,W1,...,Wn1) with
at most the free variables x1,...,xk. 35
4. Extensionality. ("z)(z Œ x ´ z Œ y) Æ (x Œ z ´ y Œ
z).
This would be trapped in interpretation power between the
existence of an (n2)subtle cardinal and an (n1)subtle
cardinal.
Presumably, again extensionality can be dropped. Also we
should be able to work with a version based on
parameterless comprehension.
PHILOSOPHY 532
PHILSOOPHCIAL PROBLEMS IN LOGIC
LECTURE 4
10/16/02
10/20/02
NOTE: The lecture on 10/16/02 is entirely superceded by
the lecture on 10/23/02. Much progress has taken place. So
these notes address a topic that was not actually covered
on 10/16/02.
1. The comparable element principle.
This work is contained in a finished manuscript "A Way Out"
on the preprint server http://www.mathpreprints.com.
The comparable element principle came about as a way out of
Russell's paradox for sets with an unexpectedly simple
escape clause.
We can think of Russell's paradox for sets in the following
way.
Informally, the full comprehension axiom scheme in the
language L(Œ) with only the binary relation symbol Œ and
no equality, is, in the context of set theory,
Every virtual set forms a set.
We use the term "virtual set" to mean a recipe that is
meant to be a set, but may be a "fake set" in the sense
that it does not form a set. The recipes considered here
are of the form {x: j}, where j is any formula in L(Œ).
We say that {x: j} forms a set iff there is a set whose 36
elements are exactly the y such that j. Here y must not be
free in j (and must be different from x). Thus {x: j} forms
a set is expressed by
($y)("x)(x Œ y ´ j).
Russell showed that
{x: x œ x} forma a set
leads to a contradiction in pure logic.
This way put of Russell's Paradox via the comparable
element principle is to modify the inconsistent Fregean
scheme in this way:
Every virtual set forms a set, or ____.
We refer to what comes after "or" as the "escape clause".
The escape clause that we use involves only the extension
of the virtual set and not its presentataion.
We are now ready to present the comprehension axiom
scheme.
\
NEWCOMP. Every virtual set forms a set, or, outside any
given set, has two inequivalent elements, where all
elements of the virtual set belonging to the first belong
to the second.
To avoid any possible ambiguity, we make the following
comments.
1. For Newcomp, we use only the language L(Œ), which does
not have equality.
2. Here "inequivalent" means "not having the same
elements".
3. The escape clause asserts that for any set y, there are
two unequal sets z,w in the extension of the virtual set,
neither in y, such that every element of z in the extension
of the virtual sets is also an element of w.
THEOREM. Newcomp is provable in ZFC + "there exists
arbitrarily large subtle cardinals". Newcomp is
interpretable in ZC + "there exists a subtle cardinal". A 37
shade less than ZFC "there exists a subtle cardinal" is
interpretable in NEWCOMP. PHILOSOPHY 532
PHILSOOPHCIAL PROBLEMS IN LOGIC
LECTURE 5
10/23/02
10/25/02
NOTE: THIS IS AN INCOMPLETE DRAFT.
We begin with the impredicative theory of types with the
axiom of choice, which accommodates a unary predicate R
whose extension forms a proper subuniverse. We augment this
theory by various relativization principles which assert
(schematically) that every sentence not mentioning R is
equivalent to the result of relativizing only certain
specified quantifiers to R. Augmentation by two such
(categories of) relativization principles of particular
simplicity results in a system which is mutually
interpretable with ZFC together with certain well studied
large cardinal axioms that are substantially stronger than
the existence of measurable cardinals. This particular
augmentation is given a reaxiomatization in more familiar
terms (elementary substructure and restricted
completeness). A substantive step towards a general theory
of relativization principles in this context is offered in
the form of an analysis of all relativization principles of
a certain form. Strong conjectures are made about
relativization principles in general.
1. Type theory with choice.
In the intended interpretation of type theory, we have
objects of type 0, classes of type 1 consisting of elements
of type 0, classes of type 2 consisting of elements of type
1, etc. We can also replace "classes" by "predicates" or
"properties".
The language of type theory, L(TT), is a many sorted
predicate calculus with a sort for each n ≥ 0. We use 38
variables xin, i,n ≥ 0, of type n. These variables range
over objects of type n.
It will be convenient to write t(x) for the type of the
variable x of L(TT).
The atomic formulas are the expressions x Œ y, where t(y) =
t(x)+1.
Formulas are built up from atomic formulas in the usual way
using the connectives ÿ,Ÿ,⁄,Æ,´, and the quantifiers ",$.
There are standard logically complete axioms and rules for
first order predicate calculus in this many sorted
language. For exposition, we will give a version here. In
16, A,B are any formulas of L(TT), and x,y are variables
of the same type.
1.
2.
3.
4.
A.
5.
B.
6. All tautologies.
"xA Æ A[x/y], where y is substitutable for x in A.
A[x/y] Æ $xA, where y is substitutable for x in A.
From A Æ B derive A Æ ("x)(B), where x is not free in
From A Æ B derive ($x)(A) Æ B, where x is not free in
From A and A Æ B, derive B. For the nonlogical axioms for type theory with choice,
below A is any formula of L(TT), and n ≥ 0.
7. Comprehension. There is a class of type n+1 whose
elements are exactly those objects of type n obeying any
given condition expressible in the language L(TT). The
sentences ($x)("y)(y Œ x ´ A), where A is a formula of
L(TT) in which x is not free.
It is standard to define equality (at each type) using all
sentences
x ≡ y ´ ("z)(x Œ z Æ y Œ z).
It is implicit in Russell that under this definition of
equality, all of the usual axioms for equality are
derivable from 17. Furthermore Russell saw the equivalence
of the above definition of equality with
x ≡ y ´ ("z)(x Œ z ´ y Œ z). 39
8. Choice. For every family of nonempty classes, where any
two classes from the family with a common element have the
same elements, there is a class containing exactly one
element from each of the nonempty classes. All sentences
(("y Œ x)($z)(z Œ y) Ÿ ("y,z Œ x)("w)((w Œ y Ÿ w Œ z) Æ
("w)(w Œ y ´ w Œ z)) Æ ($u)("y Œ x)($v)(v Œ u Ÿ v Œ y Ÿ
("w)((w Œ u Ÿ w Œ y) Æ w ≡ v)).
We call the formal system 18, TTC (type theory with
choice).
2. Type theory with choice and restriction.
In this section, we straightforwardly expand L(TT) and give
the obvious axioms that correspond to this expansion of the
language.
We only introduce a single unary predicate symbol R. The
language L(TT,R) is the language of type theory with
restriction. The atomic formulas are the expressions x Œ y,
R(z), where t(y) = t(x)+1. Formulas are built up from
atomic formulas in the usual way using the connectives
ÿ,Ÿ,⁄,Æ,´, and the quantifiers ",$.
In 16, A,B are any formulas of L(TT,R), and x,y are
variables of the same type.
1.
2.
3.
4.
A.
5.
B.
6. All tautologies.
"xA Æ A[x/y], where y is substitutable for x in A.
A[x/y] Æ $xA, where y is substitutable for x in A.
From A Æ B derive A Æ ("x)(B), where x is not free in
From A Æ B derive ($x)(A) Æ B, where x is not free in
From A and A Æ B, derive B. For the nonlogical axioms for type theory with choice,
below A is any formula of L(TT,R), and n ≥ 0.
7. Comprehension. There is a class of type n+1 whose
elements are exactly those objects of type n obeying any
given condition expressible in the language L(TT,R). The
sentences ($x)("y)(y Œ x ´ A), where A is a formula of
L(TT,R) in which x is not free.
8. Choice. For every family of nonempty classes, where any 40
two classes from the family with a common element have the
same elements, there is a class containing exactly one
element from each of the nonempty classes.
We add the axiom of restriction, that tells us that the
restriction is nontrivial.
9. Restriction. RE. R fails of some object of type 0.
($x0)(ÿR(x0)).
We call the formal system 19, TTCR (type theory with
extensionality, choice, and restriction).
A model of TTC, or TTCR, is said to be extensional if and
only if the interpretation of ≡ is identity.
Clearly TTC has an extensional model with exactly one
object of type 0, that is unique up to isomorphism. The
axiom of choice is not needed for this statement.
It is also obvious that TTCR has an extensional model with
exactly one object of type 0, and no objects of any type
that fall under R, that is unique up to isomorphism.
We close this section with a general comment about
restriction.
In the system TTCR + IR + TR that we primarily focus on
(see below), it is derivable that the class of all type
zero objects will itself fall under R. However, by
Restriction, this class has elements that do not fall under
R.
This makes sense if we think of classes as a "way" of
separating out certain objects from others. That "way" which might be viewed as an intellectual procedure  can be
in some restricted universe, where it maintains its
character but gets applied, from the point of view of the
restricted universe, only to objects in the restricted
universe.
The same considerations apply if we use "predicates" or
"properties" throughout instead of classes.
Suppose we think of sets rather classes, in the sense of
"completed totalities". Then we have a set in the
restricted universe with elements outside the restricted 41
universe. This invites serious objections in a way that
using classes does not.
Note we have relied on the well known fact that the theory
of types admits more than one closely related informal
interpretation (sets as one, and classes as another. These
correspond to notions to which Russell's paradox naturally
applies.
3. Pure Relativization.
Let A be any formula of L(TT). We construct the formula AR
of L(TT,R) by relativizing all quantifiers (Qx) to R. I.e.,
replace each
(Qx)
by
(QxR)
and expand this out to a formula of L(TT,R) in the obvious
way.
The most straightforward relativization principle is
Pure Relativization. PR. A ´ AR, where A is a sentence of
L(TT).
We consider the formal system TTCR + PR.
The usual formulation of the axiom of infinity in type
theory is the existence of a nonempty class of type 2 with
no maximal element under inclusion.
THEOREM 3.1. For each k ≥ 1, TTCR + PR proves "there exists
at least k objects of type 0". However, TTCR + PR does not
prove the axiom of infinity.
Proof: Fix k and assume in TTCR + PR that there are exactly
k objects of type 0. By PR, there must be exactly k objects
of type 0 satisfying R. From this it follows logically that
every object of type 0 falls under R. This contradicts
Restriction.
For the second claim, for each k consider the extensional
model M[k] of type theory whose type 0 objects are exactly 42
{1,...,k}. This is unique up to isomorphism. Let us fix the
finite fragment T of PR that uses exactly the sentences
A1,...,Ap from L(TT).
We claim that T has a finite extensional
each type). By obvious combinatorics, we
such that M[r] and M[k] satisfy the same
into a model M[r]* of TTCR + T by taking
to be M[k]. QED model (finite in
can find k < r
Ai's. Make M[r]
the extension of R 4. Two principles of impure relativization.
Let A be a sentence of L(TT). We will be interested in
relativizing some quantifiers in A to R and leaving other
quantifiers unchanged. When we speak of relativizing a
quantifier, we will always mean relativizing the quantifier
to R.
We now extend axiom system TTCR to include the following
impure relativization principles.
10. Initial Relativization. IR. A ´ A', where A is a
sentence of L(TT) and A' is the result of relativizing any
initial segment of quantifiers.
We will also consider the following weakening of IR.
10'. IR'. A ´ A', where A is a sentence of L(TT) and A' is
the result of relativizing any initial segment of at most 3
quantifiers.
It turns out that TTCR + IR' is enough to obtain the axiom
of infinity, and so this system get close in logical
strength to Zermelo set theory. In fact, this system (also
with IR) is equiconsistent with BZ = bounded Zermelo set
theory. When we add TRP below, the system explodes into
hefty reaches of the large cardinal hierarchy.
11. Type Relativization. TR. A ´ A'', where A is a sentence
of L(TT) and A'' is the result of relativizing all
quantifiers except the quantifiers of highest type.
We will be particularly interested in TTCR + IR + TR. We
will show that this is logically equivalent to TTCR + IR' +
TR.
5. Additional principles and logical equivalences. 43
For formulas A in L(TT), let AR be the full relativization
of A; i.e., the result of relativizing every quantifier to
R.
12. Elementary Substructure. ES. (R(x1) Ÿ ... Ÿ R(xk) Æ (A
´ AR), where A is a formula of L(TT) whose free variables
are among x1,...,xk.
13. R Completeness. RC. Every class has the same restricted
elements as some restricted class. The sentences
("x)($yR)("zR)(z Œ x ´ z Œ y).
14. All sentences ("x1,...,xkR)($yR)("z)(z Œ y ´ A) of
L(TT,R), where A is any formula of L(TT) in which y is not
free.
15. ($xR)("yR)(y Œ x ´ A), where A is any formula of
L(TT,R) in which x is not free.
LEMMA 5.1. TTCR + IR proves ES.
Proof: Suppose that we can derive ES for all formulas A of
L(TT) by induction on the formula A. This is obvious for
atomic A.
case 1. A is ÿB. Then ES for A follows immediately from ES
for B.
case 2. A is B op C, where op is ⁄,Ÿ,Æ, or ´. Then TTCR +
IRP proves
(R(x1) Ÿ ... Ÿ R(xk)) Æ (B ´ BR)
(R(x1) Ÿ ... Ÿ R(xk)) Æ (C ´ CR)
where the free variables of B are among x1,...,xk. Then TTCR
+ IR proves
(R(x1) Ÿ ... Ÿ R(xk)) Æ ((B op C) ´ (BR op CR)).
This case is completed since (B op C)R is BR op CR.
case 3. A is ($y)(B), where the free variables of A are
among x1,...,xk. By the induction hypothesis,
*) TTCR + IR proves (R(x1) Ÿ ... Ÿ R(xk) Ÿ R(y)) Æ (B ´
BR). 44
We need to show that TTCR + IR proves
(R(x1) Ÿ ... Ÿ R(xk)) Æ (($y)(B) ´ ($yR)(BR)).
Now TTCR (or even just logic) proves
("x1,...,xk)($y)(($y)(B) Æ B).
Hence TTCR + IR proves
("x1,...,xkR)($yR)(($y)(B) Æ B).
By *), TTCR + IR proves
("x1,...,xkR)($yR)(($y)(B) Æ BR).
Hence TTCR + IR proves
(R(x1) Ÿ ... Ÿ R(xk)) Æ (($y)(B) ´ ($yR)(BR))
as required.
case 4. A is ("y)(B). By induction hypothesis, we have ES
for B, and hence for ÿB. Repeating case 3 with ÿB replacing
A, we have ES for ($y)(ÿB). ES for ("y)(B) follows
immediately. QED
We wish to prove Lemma 5.1 with IRP replaced by IR'. This
requires the development of what amounts to ordered tuples
of objects, perhaps from different types.
LEMMA 5.2. Let x1,...,xk,y1,...,yk,z be distinct variables,
where t(z) = max(t(x1),...,t(xk))+2, and each t(xi) = t(yi).
There is a formula B whose free variables are exactly
x1,...,xk,z, in which y1,...,yk do not appear, such that TTCR
+ IR' proves
i) ("x1,...,xk)($z)(B);
ii) ("x1,...,xkR)($zR)(B);
iii) ("zR)(B Æ ($x1,...,xkR)(B));
iv) (B Ÿ B[x1,...,xk/y1,...,yk]) Æ (x1 ≡ y1 Ÿ ... Ÿ xk ≡ yk).
Proof: (To be polished later). The idea is to first show
that every xR has a {x} with R, and every x,yR of the
same type has an x » y with R. This shows that every
x1,...,xkR of the same type has a {x1,...,xk} with R. 45
Next, develop an ordered ktuple of distinct objects of the
same type, where if the objects have R then some ktuple
has R, and also if two k lists of distinct objects have a
common ktuple, then they are ≡. This is done by taking
<x1,...,xk> to be {{x1},{x1,x2},...,{x1,...,xk}}, which means
a set of all sets that look like one of these k terms.
Beware that we do not have extensionality, and so each of
these k terms may have many realizations, all with the same
elements. And then the whole expression also has perhaps a
multitude of realizations, all with the same elements. But
all realizations of the whole expression must have the same
elements.
Assume x1,...,xk have R. We have to prove by induction on 1
£ i £ k that some class of all representations of
{x1,...,xi} has R. We know this by IR' for i = 1. Suppose
true for i. We have to show that some class of all
representations of {x1,...,xi+1} has R. Let S be a class of
all representations of {x1,...,xi} that has R. Let S' be the
result of tacking on xi+1 to every element of S. Of course
by comprehension, some S' exists. I.e., we can write
("x)("S)($S')(S' is the result of
tacking on x to the elements of S).
By IR',
("xR)("SR)($S'R)(S' is the result of
tacking on x to the elements of S).
So this gives us the kind of ordered ktupling of objects
all of the same type that we need.
For each k, there are only finitely many "kinds" of ordered
ktuples, where "kinds" reflect the pattern of repetitions.
For each k, we name these kinds using consecutive integers
> k.
So let x1,...,xk be all of type n, with repetitions allowed.
Let p be the name of its "kind". Then the ktuple of
x1,...,xk is taken to be a class consisting of all
representations of {x1}, of {x1,x2}, ... , of {x1,...,xk},
and all classes of type n+1 with exactly p elements. Of
course, counting is with respect to ≡. It is easily verified
as above that this tupling has the required properties. 46
We still need to have tupling of objects of varying types.
The idea is to first lift them all up to the highest type
of the objects. After they are so lifted, we then apply the
tupling procedure we just developed.
Let k £ n. Objects of type k are lifted up to type n by nk
iterated singletons. Of course, we must take into account
that the singleton operation is many valued in the sense of
≡. But this does not cause problems. QED
LEMMA 5.3. TTCR + IR' proves ("x1,...,xk)($y)(A) ´
("x1,...,xkR)($yR)(A), provided the formula to the left of
´ is a sentence of L(TT).
Proof: Let B be the formula given by Lemma 5.2 using the
variables x1,...,xk,y1,...,yk,z. We can assume that
y1,...,yk,z do not appear in A. We will display all free
variables for clarity.
We claim that TTCR + IR' proves
("x1,...,xk)($y)(A(x1,...,xk,y) ´
("z)($y)(("x1,...,xk)(B(x1,...,xk,z) Æ A(x1,...,xk,y)) ´
("zR)($yR)(("x1,...,xk)(B(x1,...,xk,z) Æ A(x1,...,xk,y)).
To see this, first assume ("x1,...,xk)($y)(A(x1,...,xk,y)),
and let z be given. If ($x1,...,xk)(B(x1,...,xk,z) then let
x1,...,xk be such that B(x1,...,xk,z). Choose y such that
A(x1,...,xk,y). We have to verify that
("x1,...,xk)(B(x1,...,xk,z) Æ A(x1,...,xk,y)).
Let y1,...,yk be such that B[x1,...,xk/y1,...,yk]. By Lemma
5.2, x1 ≡ y1 Ÿ ... Ÿ xk ≡ yk. Hence A[x1,...,xk/y1,...,yk], and
our verification is complete.
Now assume ("z)($y)(("x1,...,xk)(B(x1,...,xk,z) Æ
A(x1,...,xk,y)), and let x1,...,xk be given. By Lemma 5.2,
let z be such that B(x1,...,xk,z). Let y be such that
("x1,...,xk)(B(x1,...,xk,z) Æ A(x1,...,xk,y)). Then
A(x1,...,xk,y), as required. This establishes the claim.
To complete the proof, it suffices to show that TTCR + IR'
proves
("zR)($yR)(("x1,...,xk)(B(x1,...,xk,z) Æ A(x1,...,xk,y)) ´
("x1,...,xkR)($yR)(A(x1,...,xk,y)). 47
Assume the left side, and let R(x1),...,R(xk). By Lemma 5.2,
let z be such that R(z) and B(x1,...,xk,z). Let y be such
that R(y), ("x1,...,xk)(B(x1,...,xk,z) Æ A(x1,...,xk,y)).
Then A(x1,...,xk,y).
Assume the right side, and let R(z).
case 1. ($x1,...,xk)(B(x1,...,xk,z). By Lemma 5.2, let
x1,...,xk be such that R(x1),...,R(xk),B(x1,...,xk,z). Let y
be such that R(y),A(x1,...,xk,y). Let y1,...,yk be such that
B[y1,...,yk]. By Lemma 5.2, x1 ≡ y1 Ÿ ... Ÿ xk ≡ yk. Hence
A(y1,...,yk,y).
case 2. ÿ($x1,...,xk)(Bx1,...,xk,z). Then set y to be
arbitrary such that R(y). This follows from IR'.
LEMMA 5.4. TTCR + IR' proves ES.
Proof: Examination of the proof of Lemma 5.1 reveals that
IRP' can be used throughout except for one step. This step
is
("x1,...,xk)($y)(A) ´ ("x1,...,xkR)($yR)(A)
where the left side of the biconditional is a sentence of
L(TT). I.e., we must prove this within TTCR + IR'. This is
provided by Lemma 5.3. QED
LEMMA 5.5. TTCR + TR proves RC.
Proof: We need to prove the sentences
("x)($yR)("zR)(z Œ x ´ z Œ y)
in TTCR + TR. By TTCR, it suffices to prove
*) ("u Õ R)($yR)("zR)(z Œ y ´ z Œ u)
in TTCR + TR, where u Õ R abbreviates ("x Œ u)(R(x)).
By TTCR,
("x)($y)("z)(z Œ y ´ ($w)(w Œ x Ÿ z Œ w)).
By TR, since x is the variable whose type is greater than
all other types of variables, we have 48
("x)($yR)("zR)(z Œ y ´ ($wR)(w Œ x Ÿ z Œ w)).
Let u Õ R. Let x be such that
("v)(v Œ x ´ ($b Œ u)(v ~ {b})).
Let y be such that
R(y) Ÿ ("zR)(z Œ y ´ ($wR)(w Œ x Ÿ z Œ w)).
Let z be such that R(z). Then
z Œ y ´ ($wR)(w Œ x Ÿ z Œ w).
Since we are trying to prove *), we need only
($wR)(w Œ x Ÿ z Œ w) ´ z Œ u.
For the reverse direction, let z Œ u. Then R(z). By Lemma
?, let v ~ {z}, R(v). Then z Œ v Œ x.
For the forward direction, let R(w), w Œ x, z Œ w. Let b Œ
u, w ~ {b}. Hence z ≡ b. Therefore z Œ u. QED
LEMMA 5.6. TTCR + IR' + RC proves 14 and 15.
Proof: Clearly TTCR proves all sentences
("x1,...,xk)($y)("z)(z Œ y ´ A)
of L(TT), where y is not free in A. By Lemma 5.3, TTCR +
IR' proves all sentences
("x1,...,xkR)($yR)("z)(z Œ y ´ A)
of L(TT,R), where y is not free in A.
Now let A be any formula of L(TT,R) in which x is not free.
Let x be such that ("y)(y Œ x ´ A). By Lemma 5.5, let x'
be such that R(x') and ("yR)(y Œ x ´ y Œ x'). We can
assume that x' is a variable not free in A. Then ("yR)(y Œ
x' ´ A) since x,x' are not free in A. QED
LEMMA 5.7. TTCR + ES proves IR.
Proof: We will prove the following strengthening of IR. Let
A be a formula of L(TT) with free variables among x1,...,xn, 49
n ≥ 0. Let A' be an initial relativization of A; i.e., the
result of relativizing an initial segment of the
quantifiers in A. Then
(R(x1) Ÿ ... Ÿ R(xn)) Æ (A ´ A')
is provable in TTCR + ES.
We prove this by induction on the formula A. This is
obvious for atomic A.
case 1. A is ÿB. Let A have free variables among x1,...,xn.
Let A' be an initial relativization of A. Let B' be the
corresponding initial relativization of B. Then A' is ÿB'.
By the induction hypothesis,
(R(x1) Ÿ ... Ÿ R(xn)) Æ (B ´ B')
is provable in TTCR + ES. Hence
(R(x1) Ÿ ... Ÿ R(xn)) Æ (A ´ A')
is provable in TTCR + ES.
case 2. A is B
free variables
relativization
relativization
relativization
hypothesis, op C, where op is ÿ,⁄,Ÿ.Æ, or ´. Let A have
among x1,...,xn. Let A' be an initial
of A. Let B' be the corresponding initial
of B and C' be the corresponding initial
of C. Then A' is B' op C'. By the induction (R(x1) Ÿ ... Ÿ R(xn)) Æ (B ´ B')
(R(x1) Ÿ ... Ÿ R(xn)) Æ (C ´ C')
are provable in TTCR + ES. Hence
(R(x1) Ÿ ... Ÿ R(xn)) Æ (A ´ A')
is provable in TTCR + ES.
case 3. A is ($y)(B). Let A have free variables among
x1,...,xn. Let A' be an initial relativization of A. Without
loss of generality, we may assume that the first quantifier
($y) gets relativized. Let B' be the corresponding initial
relativization of B. Then A' is ($yR)(B').
Note that the free variables of B are among x1,...,xn,y. By 50
the induction hypothesis,
*) (R(x1) Ÿ ... Ÿ R(xn) Ÿ R(y)) Æ (B ´ B')
is provable in TTCR + ES. We wish to prove
(R(x1) Ÿ ... Ÿ R(xn)) Æ (($y)(B) ´ ($yR)(B'))
in TTCR + ES. Assume R(x1) Ÿ ... Ÿ R(xn). By ES and *), we
have
($y)(B) ´ ($yR)(B*) ´ ($yR)(B) ´ ($yR)(B')
where B* is the full relativization of B.
case 4. A is ("y)(B). Repeat case 4 with $ replaced by ".
QED
LEMMA 5.8. TTCR + RC proves TR.
Proof: We will prove a stronger statement by induction on
formulas A of L(TT).
For variables x,y of the same type, we write x ~R y for
("zR)(z Œ x ´ z Œ y).
Let A be a formula of L(TT) in which the highest type of
any variable is n. Assume that the free variables of A are
among the variables x1,...,xk,y1,...,yr, where x1,...,xk have
type n and y1,...,yr have type < n. Let z1,...,zk be distinct
variables not in A, where each t(zi) = t(xi). Let A' be the
result of relativizing at least all quantifiers of type <
n. Then TTCR + RC proves
(x1 ~R z1 Ÿ ... Ÿ xk ~R zk Ÿ R(y1) Ÿ ... Ÿ R(yr)) Æ
(A ´ A'[x1,...,xk/z1,...,zk]).
QED
THEOREM 5.9. The systems TTCR + IR + TR, TTCR + IR' + TR,
TTCR + ES + RC are logically equivalent. They prove 14, 15.
Proof: By Lemmas 5.4, 5.5, we see that TTCR + IR' + TR
logically implies ES + RC. By Lemmas 5.7, 5.8, TTCR + ES +
RC logically implies IR + TR. Hence TTCR + IR' + TR
logically implies IR + TR, and so TTCR + IR' + TR and TTCR
+ IR + TR are logically equivalent. They prove 14, 15 by 51
Lemma 5.6. QED
6. Interpreting extensionality.
It is very useful to have
16. Extensionality. EXT. All sentences of the form ("z)(z Œ
x ´ z Œ y) Æ x ≡ y.
This is not provable in TTCR + IR + TR. However, we show
that TTCR + IR + TR + EXT is interpretable in TTCR + IR +
TR.
For this purpose, we define the hereditarily extensional
objects as follows. The HE objects of type 0 are just the
objects of type 0.
The HE objects of type k+1 are just the classes xk+1 such
that
i) every element of xk+1 is an HE object of type k;
ii) any object of type k that is extensionally equal to an
element of xk+1 is an element of xk+1.
The HE interpretation of any formula of L(TT,R) is obtained
by relativizing each quantifier to the HE objects of its
type, and expanding the formula out to a formula of
L(TT,R). Of course, if the original formula does not
mention R then its HE interpretation does not mention R.
We write AHE for the HE interpretation of A.
LEMMA 6.1. The HE interpretation of the closure of each
instance of EXTHE is provable in TTCR.
Proof: We show that TTCR proves
("x,yHE)(("zHE)(z Œ x ´ z Œ y) Æ
("zHE)(x Œ z ´ y Œ z)).
Assume HE(x), HE(y), ("zHE)(z Œ x ´ z Œ y), HE(z). Since
x,y consist entirely of z's with HE(z), clearly x,y are
extensionally equal. Hence x Œ z ´ y Œ z. QED
LEMMA 6.2. TTCR proves every sentence
("x1,...,xnHE)("y1,...,ynHE)(A ´ A[x1,...,xn/y1,...,yn]),
where the free variables of A are among the variables 52
x1,...,xn, A does not mention y1,...,yn, for all 1 £ i,j £ n,
xi is xj ´ yi is yj, and t(xi) = t(yi).
LEMMA 6.3. The HE interpretation of the closure of each
instance of Comprehension is provable in TTCR.
Proof: We show that TTCR proves each sentence
("x1,...,xnHE)($yHE)("zHE)(z Œ y ´ AHE)
where y is not free in A.
Let x1,...,xn have HE. Let y be such that ("z)(z Œ y ´
(HE(z) Ÿ AHE)). By Lemma 6.2, HE(y). Clearly ("zHE)(z Œ y
´ AHE). QED
LEMMA 6.4. The HE interpretation of the closure of each
instance of Choice is provable in TTCR.
Proof: It suffices to prove the following in TTCR:
Let S be an HE family of nonempty classes, where any two
classes from the family with a common element are
extensionally equal. Then there is an HE class containing
exactly one element, up to extensional equality, from each
of the nonempty classes.
Under the hypotheses, by Choice there is a class containing
exactly one element, up to ≡, from each of the nonempty
classes. This is even more than we need. QED
LEMMA 6.5. The HE interpretation of Restriction is provable
in TTCR.
Proof: Immediate. QED
LEMMA 6.6. Let A be a formula of L(TT). Then TTCR + ES
proves (AR)HE ´ (AHE)R.
Proof: We proceed by induction on A. This is obvious if A
is atomic.
case 1. A is ÿB. Then (AR)HE ´ (AHE)R is ÿ(BR)HE ´ ÿ(BHE)R.
case 2. A is B op C. Then (AR)HE ´ (AHE)R is ((BR)HE op
(CR)HE)) ´ ((BHE)R op (CHE)R). 53
case 3. A is ($y)(B). Then (AR)HE ´ (AHE)R is
($y)((HE(y) Ÿ R(y) Ÿ (BR)HE) ´
($y)((HE(y))R Ÿ R(y) Ÿ (BHE)R).
By ES, this is equivalent to
($y)(HE(y) Ÿ R(y) Ÿ (BR)HE) ´
($y)(HE(y) Ÿ R(y) Ÿ (BHE)R)
which is provable in TTCR by the induction hypothesis.
case 4. A is ("y)(B). By the induction hypothesis, the
claim holds for B, and hence for ÿB. We can repeat case 3
and obtain the claim for ($y)(ÿB). We conclude the claim
for ("y)(B). QED
LEMMA 6.7. The interpretation of the closure of each
instance of ES is provable in TTCR + ES.
Proof: Let A be a formula of L(TT) whose free variables are
among x1,...,xn. We need to show that TTCR proves
(("x1,...,xnR)(A ´ AR))HE.
This sentence is
("x1,...,xnHE)((R(x1) Ÿ ... Ÿ R(xn)) Æ (AHE ´ (AR)HE)).
By Lemma 6.6, this is provably equivalent over TTCR to
("x1,...,xnHE)((R(x1) Ÿ ... Ÿ R(xn)) Æ (AHE ´ (AHE)R)).
This follows immediately from ES. QED
LEMAM 6.8. The interpretation of each instance of RC is
provable in TTCR + ES + RC.
Proof: We have to prove
("xHE)($yRHE)("zRHE)(z Œ x ´ z Œ y)
in TTCR + ES + RC. Let HE(x). Let R(y), where x,y have the
same elements falling under R. Then
("zR)(z Œ y Æ HE(z)). 54
By ES,
("z)(z Œ y Æ HE(z)).
Also
("z,uR)((z Œ y Ÿ z,u are extensionally equal) Æ u Œ y).
This is because for z,u with R, if z Œ y and z,u are
extensionally equal, then u Œ x. Since R(u), we have u Œ y.
By IR,
("z,u)((z Œ y Ÿ z,u are extensionally equal) Æ u Œ y).
We have verified that HE(y). QED
THEOREM 6.9. TTCR + ES + RC + EXT is interpretable in TTCR
+ ES + RC.
Proof: By Lemmas 6.1, 6.3, 6.4, 6.5, 6.7, 6.8. QED
7. Sketch of proof that TTCR + ES + RC is very strong.
We will show that ZFM = ZFC + "there exists a measurable
cardinal" is interpretable in TTCR + ES + RC, or
alternatively, in TTCR + IR + TR. By Theorems 5.9 and 6.9,
it suffices to interpret ZFM in 116.
A linear ordering of type n objects consists of a class of
type n+2 which is linearly ordered under inclusion, and
where every object of type n lies in some element of the
linear ordering. One type n object is less than another in
the sense of the linear ordering if and only if the first
lies in some element of the linear ordering that the second
does not lie in.
A well ordering of type n objects is a linear ordering of
type n objects where every nonempty set of type n objects
has a least element.
LEMMA 7.1. 116 proves that there is a well ordering of the
objects of any given type.
Proof: (to be polished later). Adapt the Zermelo well
ordering theorem to this context. QED 55
THEOREM 7.2. It is provable in 116 that there is a
countably additive nonprincipal ultrafilter on all subsets
of R[0].
Proof: Let W be a well ordering of the class 0 objects. By
IR, we can assume that R(W).
Suppose W does not have a limit point. Then it is easy to
prove that every point falls under R. Hence by R, W has a
limit point. Therefore it has a least limit point, and the
initial segment up to there will serve as the natural
numbers. One also sees that that initial segment also falls
under R, and so it also serves as the natural numbers in
the sense of R. Full induction and recursion is supported
both in the usual sense and in the restricted universe
represented by R.
Let R[0] = {x0: R(x0)}. Fix c = c0 with ÿR(c). Let X Õ R[0].
By RC, let R(X*), where X* and X have the same elements
falling under R. By ES, X* is unique.
Let K = {X Õ R[0]: c Œ X*}. Note that if X is a singleton
then X* = X. Obviously K is an ultrafilter on the subsets
of R[0], where singletons lie outside K.
We will show that K is a countably complete ultrafilter on
the subsets of R[0].
Let X0,X1,... be an infinite sequence of elements of K
indexed by the natural numbers in the sense of the second
paragraph. So c lies in X0*,X1*,... .
We claim that R({X0*,X1*,...}). To see this, it suffices to
prove that Y = {{{0},X0*},{{1},X1*},...} falls under R.
First note that each {i} falls under R. Since each Xi* falls
under R, we see that each {{i},Xi*} falls under R.
Let X be the intersection of the Xi's. We claim that X* is
the intersection of the Xi*'s. To see this, first note that
X* and the intersection of the Xi*'s fall under R. The
former is by the definition of the * operator, and the
latter is from the fact that Y falls under R, and IR.
Hence to see that X* is the intersection of the Xi*'s, it
suffices to show that the two sets have the same elements
that fall under R. Now the elements of X* that fall under R 56
are precisely the elements of X. So it suffices to show
that the elements common to all Xi*'s, that fall under R,
are exactly the elements of X.
Let x Œ X. Since each Xi includes X, we see that each Xi*
includes X*, and so x lies in each Xi*.
On the other hand, suppose x lies in each Xi* and R(x). By
the definition of the * operator, x lies in each Xi. Hence x
Œ X.
We have now established that X* is the intersection of the
Xi*'s. Hence c Œ X*. Therefore X Œ K, as required. QED
THEOREM 7.3. The following is provable in PA. If TTRC + ES
+ RC (or TTRC + IR + TR) is consistent then so is BZCM =
BZC + "there exists a measurable cardinality".
Let ZC is Zermelo set theory with the axiom of choice, and
BZC is the slightly weaker system where separation is
restricted to formulas with bounded quantifiers. Also, we
write "measurable cardinality" because without replacement
in ZC, we cannot get a good theory of ordinals and
cardinals. But nevertheless, we can well order any set in
BZC, and so "measurable cardinality" carries a great deal
of force. E.g., by standard arguments, ZFC + "there exists
a Ramsey cardinal" is interpretable in BZC + "there exists
a measurable cardinality".
However, we can go much further. But first we will go just
a bit further to ZFM as follows.
THEOREM 7.4. ZFM is interpretable in TTRC + ES + RC (or
TTRC + IR + TR).
Proof: Look at the previous argument where c is chosen to
be the least element not in R[0] in the well ordering W
with R(W). One can go further and see that c is in fact a
measurable cardinal, by checking < c additivity of the
ultrafilter K.
Now let d (as a point in W) be the least measurable
cardinal. Since d is definable from W, and W falls under R,
we see that d falls under R. We have thus shown that there
are at least two measurable cardinals. In fact, we get much
more than ZFC + there exists a measurable cardinal out of
this. QED 57
We now show how to obtain an interpretation of ZFC + {there
is an elementary embedding from some V(k+n) into some
V(l+n), k < l}n in TTRC + ES + RC (or TTRC + IR + TR).
NOTE: TO BE COMPLETED LATER.
8. Interpreting 116.
Let j:V(k+w) Æ V(l+w) be a nontrivial elementary embedding
with critical point b < k. We take the set of type 0 objects
of U to be V(l), and build up from there using V(l+w). We
take the objects of R to be the values of j.
The only delicate point is the verification of RC. This
amounts to verifying the following. Let S be a subset of
j[V(k+n)]. Then there exists x Œ rng(j) such that x,S have
the same elements from rng(j).
Fix S Õ j[V(k+n]) and write S = j[S'], S' Õ V(k+n). Note
that for all y Œ V(k+n),
j(y) Œ j(S') ´ y Œ S' ´ j(y) Œ S.
Hence j(S') and S have the same elements from j[V(k+n)], and
hence the same elements from rng(j). QED
9. The general theory of relativization.
We use a language LTQ (language of typed quantifiers) for
discussing the quantifiers in arbitrary sentences of L(TT).
We write p(Q,A) for the position of the quantifier Q in the
sentence A.
We write t(Q,A) for the position of the type of the
quantifier Q in the sentence A in the list of types of
quantifiers in A in decreasing order. Thus the quantifiers
Q of highest type in A have t(Q,A) = 1.
The atomic formulas in LTQ are
p(Q) = i
t(Q) = i
where i ≥ 1. Here Q is the unique variable of LTQ and each i 58
≥ 1 is treated as a constant rather than a variable.
Formulas in LTQ are built up from the atomic formulas of
LTQ using the usual connectives ÿ,Ÿ,⁄,Æ,´.
Note that if A has exactly k quantifiers and i > k, then
p(Q) = i will be satisfied by no quantifier. Also if A has
exactly k sorts and i > k, then t(Q) = i will be satisfied
by no quantifier.
A relativization rule is just a formula a in LTQ.
Relativization rules a are intended to apply to sentences A
of L(TT), resulting in a sentence a(A) of L(TT,R). The idea
is that a(A) is the result of relativizing exactly those
quantifiers Q in A that obey a.
A relativization principle is again given by a
relativization rule a, and is identified with the scheme of
all equivalences
A ´ a(A)
where A is a sentence of L(TT).
Note that axiom IR' can be construed as three separate
relativization principles given by the formulas
p(Q) = 1 Ÿ ÿp(Q) = 1 (relativize no quantifiers)
p(Q) = 1 (relativize only the first quantifier)
p(Q) = 1 ⁄ p(Q) = 2 (relativize only the first two
quantifiers)
p(Q) = 1 ⁄ p(Q) = 2 ⁄ p(Q) = 3 (relativize only the first
three quantifiers).
Note that axiom IR can be construed as the infinitely many
relativization principles given by the formulas
p(Q) = 1 ⁄ ... ⁄ p(Q) = k
where k ≥ 0. The empty disjunction corresponds to truth,
which means that no quantifiers are relativized.
Note that axiom TR can be construed as the single
relativization principle given by the formula
t(Q) = 1. 59
THEOREM 8.1. Let a be a relativization principle (given by
a formula in LTQ). Then TTCR + PR + a is consistent if and
only if 116 proves a.
THEOREM 8.2. 116 is logically equivalent to TTCR together
with the union of all relativization principles (given by a
formula in LTQ) which are consistent with TTCR + PR.
10. Removing choice.
We are working on a version without choice. The axiom of
restriction must be strengthened. The basic idea is that
any "large" class must have elements that do not fall under
R. The interpretation power will be roughly the same as
above, or at least as large as measurable cardinals of high
order.
11. Towards very general contexts.
Suppose we start with a theory (with no constant or
function symbols) which is axiomatized by schemes involving
all formulas in its language. We can then add a unary
predicate symbol R, and restate the axioms where the
schemes are now extended to all formulas in the expanded
language. We then add a principle of restriction  either
that there is at least one object that does not fall under
R, or perhaps that the "large" extensions of formulas have
at least one object that does not fall under R. Then we
make a study of the relativization of sentences. The idea
is that in many contexts, perhaps one gets enormous logical
strength out of this.
PHILOSOPHY 532
PHILOSOPHICAL PROBLEMS IN LOGIC
LECTURE 6
11/6/2002
11/7/2002
NOTE: We talked about a simplifying approach via the
cumulative theory of types, but I don't think this works.
Also we talked about subjective isomorphisms, and this is a
big topic with big potential which I am currently working
out. So that should appear in some supplementary notes
later.
************* 60
We discuss a new approach, related to Lecture 5, for
getting at what’s philosophically behind set theory with
large cardinals. We comment on the relationship between
this and the previous work.
The previous work had two aspects. A first is the
relationship between a type structure and a type
substructure, which leads to reasonable axioms concerning
their interaction. The intention is to see the
philosophical plausibility and/or clarity of what is being
asserted.
The second is that after writing down the obvious axioms
for a type structure and a type substructure, not including
the two axiom schemes that drive the first aspect, we can
make a general study of relativization of quantifiers in
sentences about the type structure to the type
substructure. Partial results support the conjecture that
one can completely analyze all syntactic relativization
rules and show that this leads to an array of formal
systems that line up linearly with a highest level
logically equivalent to what we obtain in the previous
paragraph, which corresponds to roughly extendible
cardinals. One also has support for the conjecture that if
a set of relativization rules are applied, either they lead
to “obvious” inconsistencies, or they are consistent.
We think that the new approach below seems arguably more
compelling and fundamental than the first aspect above.
However, it does not seem to be any kind of direct
competitor for the second aspect above.
1. Type theory.
We begin with the theory of types, TT, with variables over
each sort 0,1,2,..., and where the atomic formulas are of
the form
xn Œ yn+1.
The sole axioms are the full comprehension axioms, which
are formulas in the language of the form
($x)("y)(y Œ x ´ j)
where j is a formula in the language in which x is not
free. 61
This system has an obvious model with exactly one object of
type 0, and finitely many objects at each type.
We now consider the following principle. To formalize it in
TT we rely on well known methods of formalization in TT.
Call a set at any nonzero type large if and only if it
cannot be mapped oneone into type 0 objects. The
formalization of this concept in TT is well known using
Leibniz equality
x ≡ y ´ ("z)(x Œ z ´ y Œ z)
and ordered pairing across types. Ordered pairing across
types is achieved by leveling out the types using iterated
singletons.
In TT, binary relations are treated as sets of ordered
pairs of objects of a given type. The ordered pair <x,y> is
taken to be {{x},{x,y}}, where there are possibly many such
ordered pairs. In fact there are possibly many {x}. All of
them have exactly one element, namely x, and this is
formulated using Leibniz equality.
A relation on type n is a set of ordered pairs of objects
of type n. Its field may be any set of objects of type n. A
relation on type n is itself of type n+3.
Here is the statement of the first principle.
T1. Every large relation on type n is second order (higher
order) equivalent to a proper subset.
For each n ≥ 1 this is a statement or scheme. For second
order, or any specific level of order, we can either
formalize T1 for n as a single sentence using a truth
definition for second order logic, or any specific level of
order, available in TT, or formalize it as a scheme.
The most straighforward formalization of T1 is as a double
scheme, indexed by the choice of n and by the choice of
finitely many sentences in higher order logic that are to
be transferred. This does not require the construction of
any truth predicates within T1.
There are really 3 different kinds of formalization of T1. 62
i) use truth definitions as much as possible. Still, we
cannot consolidate the types n nor can we consolidate the
levels of order (in higher order);
ii) use schemes entirely, avoiding any truth definitions,
but only for choices of single higher order sentences to be
transferred.
iii) use schemes entirely, avoiding any truth definitions,
and use arbitrary finite sets of higher order sentences to
be transferred.
THEOREM 1.1. All three ways of formalizing T1 are logically
equivalent. T1 is mutually interpretable and equiconsistent
with ZFC + {there exists an nextendible cardinal}n.
The calculation of specific n's and levels of order that
give high strength is unclear. We are confident that n = 3
and second order, using schemes throughout without truth
definitions, gives more than a measurable cardinal in
strength, and at least something close to a 1extendible
cardinal in strength.
As we shall see in section 5, the axiomatization is
simplified considerably in the cumulative theory of types
because the notion of largeness can be avoided completely.
2. Type theory with choice.
Relations whose field is the set of all type 0 objects are
of particular importance, and we wish to accomplish what we
did in section 1 with such relations. We call such
relations
full relations on type 0.
In section 3 we consider full relations on type n.
We will be using the axiom of choice in the following form,
in every type.
Let x be a set of nonempty sets, any two of which
have the same elements or no elements in common.
There exists a set which has exactly one element
in common with each element of x. 63
Here “exactly one” is formulated in terms of Leibniz
equality. Also x is of any particular type ≥ 2.
This results in
We consider the
remarks about 3
section 1 apply the system TTC (type theory with choice).
following second order principles. The
basic choices of formalization made in
here as well. It seems clear, but we have not checked the details, that
TT plus any or all of the principles T2  T7 below should
be no stronger than TT plus the axiom of infinity. So
choice is really needed.
T2. Every full relation on type 0 is second order (higher
order) equivalent to a proper subset.
THEOREM 2.1. TTC + T2 corresponds roughly to indescribable
cardinals. At higher types, we still do not go beyond such
cardinals.
T3. Every full relation on type 0 is second order (higher
order) equivalent to two proper subsets, neither of which
is a subset of the other.
THEOREM 2.2. TTC + T3 corresponds roughly to a 1extendible
cardinal.
T4. Every full relation on type 0 is first order equivalent
to a proper subset of the same cardinality.
THEOREM 2.3. TTC + T4 interprets sharps and is
interpretable in a Ramsey cardinal.
This is because T4 is intimately related to Jonsson
cardinals in set theory.
T5. Every full relation on type 0 is second order (higher
order) equivalent to a proper subset of the same
cardinality.
THEOREM 2.4. TTC + T5 corresponds roughly to a nontrivial
elementary embedding from V into M, where M is a transitive
class containing the power set of the first fixed point
above the critical point, in the case of second order. For
higher order, roughly L(V(k)) into itself, which is below
V(k+1) into itself. 64
A nontrivial embedding from some V(k+1) into itself is the
strongest large cardinal axiom normally considered that is
believed to be consistent with the axiom of choice. A bit
stronger and more technical is L(V(k+1)) into itself.
T6. Every full relation on type 0 is isomorphic to a proper
subset.
THEOREM 2.5. T6 is inconsistent.
3. Type theory with extensionality and choice.
T7. Every full relation on type n is second order (higher
order) equivalent to a proper subset.
THEOREM 3.1. TTEC + T7 for n = 0 corresponds roughly to an
indescribable cardinal. For n ≥ 1 it corresponds roughly to
an nextendible cardinal.
4. Type theory with indistinguishability.
We now discuss principles of indistinguishability in type
theory. The indistinguishability of relations is stronger
than their higher order equivalence. We generally obtain
greater logical strength using indistinguishability rather
than higher order logical equivalence.
Let x,y be of the same type. We say that they are
indistinguishable if and only if they obey the same
properties with no parameters. This is formulated as a
scheme.
Recall the notion of large in TT. A full relation on A is a
binary relation whose field is A.
In TT, we consider
T8. In each large cardinality and type there is a set A
such that every full relation on A is indistinguishable
from a proper subset (superset).
This can be formalized with finite conjunctions of
transfer, or with single property transfer. They are
provably equivalent. Also it is provably equivalent to
formalizations that use truth predicates available in TT.
THEOREM 4.1. TT + T8 interprets extendible cardinals and is 65
interpretable in Vopenka’s principle.
In TTC, we consider
T9. There is a set A of type 0 objects such that every
relation with field A is indistinguishable from a proper
subset (proper superset).
THEOREM 4.2. TTC + T9 corresponds roughly to a subtle
cardinal. These are stronger than indescribable cardinals.
T10. There is a set A of type 0 objects such that every
relation with field A is indistinguishable from two proper
subsets (proper supersets), neither of which is a subset of
the other.
THEOREM 4.3. TTC + T10 has interpretation power between an
extendible cardinal and Vopenka's principle.
T11. There is a set A of type 0 objects such that every
relation with field A is indistinguishable from two
isomorphic proper subsets (proper supersets), one of which
is a proper subset of the other.
THEOREM 4.4. TTC + T11 corresponds to roughly a nontrivial
elementary embedding from a rank into itself.
PHILOSOPY 532
PHILOSOPHICAL PROBLEMS IN LOGIC
LECTURE 7
11/13/02
11/14/02
We now take up our second topic: logic in the universal
domain.
1. General remarks.
Frege intended his quantifiers to range over absolutely
everything. An obvious question is:
if the quantifiers are to range over
absolutely everything in first order
predicate calculus then what sentences
are satisfiable?
The interpretation of predicate calculus in any domain 66
involves constants, multivariate relations, and
multivariate functions on that domain. When the domain is a
set, this is a matter of set theory, and is well
understood. Even here, there are some interesting delicate
issues, as we shall see.
But when the domain is something outside the realm of set
theory, and outside the realm of mathematics, then we need
to think carefully about the nature of multivariate
relations and functions on that domain.
It could be the case that the answer to our question is
greatly dependent on the precise nature of the multivariate
relations and functions on the universal domain. However,
we shall see that only relatively modest  and plausible principles about the relations and functions on the
universal domain are sufficient to determine the answer to
our question. For relatively modest fragments of predicate
calculus, the principles needed are relatively modest. As
we bite off more and more of predicate calculus, we will
need sharper principles.
In this regard, the situation is not all that different
from the usual set theoretic interpretation of predicate
calculus (in set theoretic domains). Here, very little is
needed about the nature of the multivariate relations and
functions on the domain in order to determine the sentences
satisfiable in that domain.
The relevant principles about the multivariate relations
and functions on the universal domain generally assert that
every multivariate relation acts very symmetrically on some
(usually finite) list of distinct objects. Functions will
throughout be treated as univalent relations. Thus the
relevant principles are principles of indiscernibility.
2. Official presentation of predicate calculus.
We care about full predicate calculus and also its
fragments. In full predicate calculus we have
i) variables xn, n ≥ 1;
ii) constant symbols cn, n ≥ 1;
iii) relation symbols Rnm, n,m ≥ 1. The arity is n;
iv) function symbols Fnm, n,m ≥ 1. The arity is n;
v) connectives ÿ,Ÿ,⁄,Æ,´;
vi) quantifiers ",$; 67
vii) equality =.
The terms are given by the following inductive clauses:
i) every variable and constant is a term;
ii) if F is a function symbol of arity n and t1,...,tn are
terms, then F(t1,...,tn) is a term.
The atomic formulas are the expressions
s=t
R(t1,...,tn)
where s,t,t1,...,tn are terms, and R is an nary relation
symbol.
The formulas are given by the following inductive clauses:
i) every atomic formula is a formula;
ii) if j,y are formulas then so are (ÿj), (j ⁄ y), (j Ÿ y),
(j Æ y), (j ´ y).
iii) if j is a formula then ("xn)(j), ($xn)(j) are formulas.
We call the above language, PC(=). The semantics of PC(=)
are well known, using any nonempty domain of objects. Of
course, this involves the concepts of multivariate
relation/function on that domain. The semantics of PC(=)
was first given formally by Tarski.
We are interested in fragments of PC(=). A particularly
important way of specifying fragments is in terms of giving
a set of the constant, relation, and function symbols, and
also asserting whether or not = is allowed. The constant,
relation, and function symbols are called the nonlogical
symbols.
Let s be a set of nonlogical symbols. We write PC(s) for
predicate calculus using only nonlogical symbols from s. We
write PC(s,=) for predicate calculus using only nonlogical
symbols from s and =.
Sometimes constant and function symbols are awkward, and
we use the important fragments PC(rel) and PC(rel,=). Here
"rel" is the set of all relation symbols.
Another independent way of designating fragments of PC(=)
is to specify a quantifier prefix for all formulas to be 68
considered. E.g., such expressions as PC(s,"..."),
PC(s,=,"..."$...$) are self explanatory.
3. Set domains.
We now take up the case where the domain is a set. This is
definitely not the case of the universal domain:
THEOREM 3.1. No set has everything in it. In particular, no
set has all sets in it.
Proof: Let S be a set. Consider the set x = {y Œ S: y œ y}.
Since S is a set, x is a set. If x Œ S then x Œ x ´ x œ x
which is a contradiction. QED
THEOREM 3.2. A set of sentences of PC(=) is satisfiable (in
some nonempty set) iff it is consistent within any one of
the many standard systems of axioms and rules of inference
for PC(=).
How can this be established conveniently with a minimum of
commitment? Since the domains are sets, it is natural for
us to formulate these Theorems in set theory. Let BST'
(basic set theory prime) be the following system in Œ,=:
1. Extensionality.
2. Pairing.
3. Union.
4. The set of all finite subsets of any set exists.
5. Separation for bounded formulas.
6. Infinity. There is a least set x such that ∅ Œ x and ("y
Œ x)(y » {y} Œ x).
7. Induction for all formulas.
THEOREM 3.3. Theorem 3.2 is conveniently formalizable and
is provable in BST'.
Proof: This is enough to conveniently handle the syntax of
PC(=) and the axioms and rules of inference. We can also
conveniently handle the finite tuples from x of any
indefinite length. This supports multivariate relations and
functions. Thus we have structures of the kind that are the
interpretations of PC(=).
We have to formalize the notion of truth of a sentence of
PC(=) in a structure with a given nonempty set domain. We
use the variant of Tarksi's truth definition, which 69
involves partial satisfaction relations. These partial
satisfaction relations exist of any indefinite level by a
basic induction argument.
The forward direction is by a well known induction. We show
that in every proof in PC(=) from a set of sentences of
PC(=), every line is true in any given model of that set of
sentences. Hence a satisfiable set of sentences of PC(=)
must be consistent.
We now come to the reverse direction. Here is where matters
could get quite delicate but they do not. The usual proofs
of completeness start with a consistent set S of sentences
in PC(=) and find a model of S with domain N = the set of
all nonnegative integers, but with = interpreted as an
equivalence relation. In fact, one actually constructs a
bit more than is needed: the entire satisfaction relation
for the structure. This construction can be easily carried
out in BST'. Then the factoring out by the equivalence
relation is readily carried out in BST'. This results in a
model whose domain is a nonempty initial segment of N
(possibly all of N). We will also have the entire
satisfaction relation for the structure. QED
Note we skirted all philosophical issues in establishing
Theorem 3.2. This is provided one accepts BST', with its
minimum set theoretic commitments. This is because the
domains that are the proper initial segments of N are
sufficient, and the multivariate relations/functions on
them needed for satisfying consistent theories are so
directly definable from the theories.
We now consider a more delicate matter. Let us be given a
nonempty set D, which will serve as the domain of our
interpretations of PC(=). What can we say about the sets of
sentences of PC(=) that are Dsatisfiable; i.e., have a
model with domain D?
If D is finite, then of course there are no philosophical
conundrums involved. However, what if D is infinite?
INF is the set of sentences
($x1)...($xn)(x1 ≠ ... ≠ xn).
DIGRESSION. This sentence has length quadratic in n.
However, the logically equivalent reaxiomatization 70
("x1)...("xn1)($xn)(xn ≠ x1 Ÿ ... Ÿ xn ≠ xn1)
has length linear in n.
THEOREM 3.4. The following is provable in BST'. A set of
sentences in PC(=) is Nsatisfiable iff it is consistent
with INF. If a set of sentences in PC(=) is Esatisfiable
for finite E of infinitely many different sizes, then it is
Nsatisfiable.
Let BST* be
1. Extensionality.
2. Pairing.
3. Union.
4. The set of all finite subsets from any set exists.
5*. Separation for all formulas.
6. Infinity.
A well known generalization of the completeness theorem,
normally proved using the compactness theorem for
arbitrary sets of sentences, will establish the following.
THEOREM 3.5. The following is provable in BST*. Let D be an
infinite well ordered set. A set of sentences in PC(=) is
Dsatisfiable if and only if it is Nsatisfiable if and
only if it is consistent with INF.
From the modern set theoretic point of view, every set is
well ordered. Hence Theorem 3.5 disposes of the question of
characterizing the Dsatisfiable sets of sentences in
PC(=). If D is finite then there is nothing much to say
(although interesting from the point of view of
computational complexity), and the Dsatisfiable sets of
sentences in PC(=) are all incomparable under inclusion as
the size varies. On the other hand, if D is infinite, then
the Dsatisfiable sets of sentences in PC(=) are all the
same and different from any of the finite domain cases.
But we are interested in non set theoretic domains, and in
particular the universal domain. We maintain that there is
no good reason to think that all domains are set theoretic
in character, at least in the sense that they have a well
ordering.
In fact, let us recall just why set theorists believe that 71
any set is well ordered. This is the Zermelo well ordering
theorem. To prove it, we basically need a choice function
that picks an element out of every nonempty subset.
But which choice function? In set theory, we just take its
existence as part of the setup of set theory. It is now
standard to do this, armed with the knowledge that it
facilitates a lot of elegant abstract mathematics. Another
rationale is to say that it is inherent in the very concept
of set that any imaginable arrangement for a set is
realized. And a choice function  as a set theoretic object
 is just such an imaginable arrangement.
However, we can either take the concept "domain" to be
quite different, where we require explicitness, or at least
some semblance of explicitness for carving out relations
and functions on the domain. E.g., this may have to do with
some inherent indiscernibility of the elements of the
domain that one is compelled to respect. Or,
alternatively, we can say that we are interested in
interpreting PC(=) by means of explicit predicates and
functions.
In any case, it would seem that no matter how we talk about
this, we must adopt such a view about the universal domain
if we are going to adopt such a view about any domain.
E.g., if we can well order the universal domain then we can
clearly well order any domain. Even if we can linearly
order the universal domain then we can linearly order any
domain. So it seems reasonable to take as inherent in the
universal domain that it cannot be well ordered, or even
linearly ordered.
If the universal domain does not have a linear ordering,
then of course we know that the sentence
R is a linear ordering
is not satisfiable, and so we are in for some work to
figure out what the satisfiable (sets of) sentences are.
Clearly the universal domain is not behaving like any
ordinary infinite set like N.
Is there an interesting threshold for richness of the
multivariate relations on D so that Dsatisfiability is the
same as Nsatisfiability? 72
A beautiful one. Let us call D logically complete iff for
all sets of sentences in PC(=), Dsatisfiability is the
same as Nsatisfiability.
Let FS(D) be the set of all finite sequences from D. Finite
sequences are defined as functions from proper initial
segments of w, which are defined in terms of ordered pairs,
which are in turn defined in terms of unordered pairs. Note
that FS(D) can be proved to exist in BST'.
LEMMA 3.6. The following is provable in BST'. Let F:DxD Æ D
be oneone, where D has at least two elements. Then F has a
countably infinite subset. Furthermore, there is a oneone
G:FS(D) Æ D.
Proof: Let F:DxD Æ D be oneone. Fix x1,x2 to be two
distinct elements of D. Since F is oneone, there are four
values of F at arguments x1,x2. Set x3 to be the first one
that is not x1,x2. Now there are nine values of F at
arguments x1,x2,x3. Set x4 to be the first one that is not
x1,x2,x3. Continue in this way. This generates a countably
infinite subset of D.
Let x Œ FS(D). Write x = <x1,...,xn> and prove the existence
of the finite sequence y1,...,yn where y1 = x1, y2 = x2, and
for i ≥ 3, yi = F(yi2,yi1). Let z = <z1,...,zn> and w1,...,wn
be the corresponding finite sequence. It is clear that for
each i, yi = wi if and only if x1 = z1 Ÿ ... Ÿ xi = zi. Hence
xn = zn ´ x = y. Thus we define Gn(x) = yn, where Gn maps
sequences from D of length n into D. Then Gn is oneone. By
the first claim, we have a oneone function H:wxD Æ D. So
define G:FS(D) Æ D by G(x) = H(n,Gn(x)), where x has length
n. QED
We refer to a oneone function F:DxD Æ D as a pairing
function.
LEMMA 3.7. The following is provable in BST'. Let D be a
linearly ordered set with a pairing function and at least
two elements. Let T be a set of sentences in PC(=)
consistent with INF. Then T has a weak model whose domain
is the set of closed terms in the extension of PC(=) by
Skolem function symbols and the constants cx, x Œ D. The
equality relation between closed terms s,t depends only on
i) the symbols in s = t from left to right with the cx all
replaced by a common marker, and ii) the order type of the
subscripts in the cx's that appear in s = t from left to 73
right.
Proof: Let < be a linear ordering of D and F:DxDÆ D be oneone. It will be convenient to make (D,<) into a dense
linear ordering (D,<') without endpoints, by surrounding
each point with a copy of the rationals. To make this
construction in BST', first consider the lexicographic
order on DxQ using the < on D. By Lemma 3.6, we can convert
the domain to D.
Suppose T is a set of sentences consistent with INF. By
standard model theory, doable in BST', T has a model M
(with domain N) with countably infinitely many Skolem
functions, where every element of the domain is generated
by these Skolem functions over an infinite set of linearly
ordered indiscernibles of order type Q.
Note that if a model satisfies every universal sentence
that M satisfies, then that model satisfies T. This is a
crucial fact about Skolemization.
We now construct a term model M* based on M, which will be
seen to satisfy every universal sentence that M satisfies.
Introduce the new constants cx for each x Œ D. We define a
structure M* whose domain S consists of the closed terms in
these constants and the constant and function symbols of M.
We will use the linear ordering of D, which linearly orders
the subscripts of the new constants.
First let c be a constant in L(M). The interpretation of c
in M* is the closed term c, which is an element of S.
Now let F be a kary function symbol of M. Let t1,...,tk Œ
S. Take F*(t1,...,tk) to be the closed term F(t1,...,tk).
Finally, let R be a kary relation symbol of M. We compute
the truth value of R*(t1,...,tk) by first making any order
preserving assignment of indiscernibles in M to the new
constants appearing in the t1,...,tk. This results in terms
t1',...,tk' with parameters from the indiscernibles in M.
The truth value of R*(t1,...,tk) is taken to be the truth
value of R(t1',...,tk') in M. Because of indiscernibility in
M, this calculation is independent of the choice of the
order preserving assignment of indiscernibles to the new
constants appearing in the t1,...,tk. 74
The above paragraph is also done for the special 2ary
relation symbol =. Note that the interpretation in M* of =
is a binary relation, but not necessarily equality.
Suppose that a sentence j(t1,...,tn) holds in M*, where j is
a quantifier free formula and t1,...,tn Œ S. By making an
order preserving assignment of indiscernibles in M to the
new constants appearing in t1,...,tn, we see that
j(t1',...,tn') holds in M*. This is because the truth values
of the constituent atomic subformulas of j(t1,...,tn) are
preserved. This establishes that every universal sentence
true in M remains true in M*. By the Skolemization, we now
see that M and M* satisfy the same sentences of PC(=), and
hence M* satisfies T. QED
We will refer to the equivalence relation given by Lemma
3.7 that is the interpretation of = in M* by ≡.
At this point, it is standard to simply take M* factored
out by ≡ in Lemma 3.7 in order to get a model of T. However,
we want to stay within BST' and be much more explicit. We
also have an eye to dealing with the universal domain
later, in which case also factoring out by the equivalence
relation is not acceptable.
What we will use is a canonical presentation of definable
relations in any dense linear ordering without endpoints.
This is of some independent interest.
Let (D,<) be a dense linear ordering without endpoints.
Let j(x1,...,xk,b1,...,bp) be a formula in <,=, where all
free variables are shown, and there are parameters b1 < ...
< bp from D. We view j as defining a kary relation on D
carved out by the distinct variables x1,...,xk.
LEMMA 3.8. Suppose there exists b1' < b2 such that
j(x1,...,xk,b1,...,bp) and j(x1,...,xk,b1',b2,...,bp) define
the same relation. Then for all b1' < b2,
j(x1,...,xk,b1,...,bp) and j(x1,...,xk,b1',b2,...,bp) define
the same relation.
Proof: Assume hypotheses. This states a definable property
of b1,...,bp that is shared by b1',b2,...,bp, and the
definable property has no parameters. By quantifier
elimination, falling under any given definable property
without parameters is just a matter of order type. QED 75
LEMMA 3.9. Suppose that one of the parameters b1,...,bp can
be moved, where the parameters still are strictly
increasing, so that j(x1,...,xk,b1,...,bp) still defines the
same relation, as in Lemma 1 for b1. Then that parameter can
be so moved to any alternative spot with the same result,
as in Lemma 1.
Proof: Same as for Lemma 3.8. QED
LEMMA 3.10. Assume the hypotheses of Lemma 3.0. Then there
is a formula y(x1,...,xk,b1,...,bi1,bi+1,...,bp) that defines
the same relation as j(x1,...,xk,b1,...,bp).
Proof: Let y(x1,...,xk,b1,...,bi1,bi+1,...,bp) be
($bi)(j(x1,...,xk,b1,...,bp) Ÿ bi1 < bi < bi+1). QED
LEMMA 3.11. Let j(x1,...,xk,b1,...,bp) and
y(x1,...,xk,c1,...,cp) define the same relation, b1 < ... <
bp, and c1 < ... < cp, and {b1,...,bp} ≠ {c1,...,cp}. Then
there exists r(x1,...,xk,b1,...,bi1,bi+1,...,bp), some i, or
there exists r(x1,...,xk,c1,...,cj1,cj+1,...,cp), some i,
that defines the same relation.
Proof: The hypothesis is a definable property of
(b1,...,bp,c1,...,cp), without parameters. Let i be least
such that bi ≠ ci. If bi < ci. Then we can move bi around so
that Lemma 3.10 holds. If ci < bi then we can move ci around
so that Lemma 3.10 holds. QED
Here is the model theoretic fact of some independent
interest.
LEMMA 3.12. Let R be a kary definable relation in (D,<)
which can be defined by a formula with p parameters and no
fewer. Then all formulas with p parameters that define R
have exactly the same parameters.
Proof: This is immediate from Lemma 3.11. QED
We say that D is logically normal if and only if for any
set T of sentences in PC(=), T is Dsatisfiable if and only
if T + INF is consistent.
THEOREM 3.13. The following is provable in BST'. D is
logically normal if and only if i) D has at least two
elements; ii) D has a linear ordering; and iii) there is a 76
oneone function from DxD into D.
Proof: Since "R is a linear ordering", "binary F is oneone", and "there are at least two elements" are each
consistent with INF, we see that if D is logically normal
then i), ii), iii) hold. This establishes the forward
direction.
For the reverse direction, assume i)  iii) and apply Lemma
3.7. We obtain a model M* of T whose domain is the set of
closed terms in the extension of PC(=) by infinitely many
Skolem function symbols and the constants cx, x Œ D. In
addition, we have the condition expressed in the last
sentence of Lemma 3.7.
Normally, we complete the proof by factoring out by this
equivalence relation.
We would like to avoid doing this, as we will be interested
in carrying out this argument for domains of unbounded
extent, like the universal domain. The problem here is that
the ensemble of equivalence classes moves us up a type.
We wish to develop a suitable canonical presentation of
each equivalence class under the equality relation among
closed terms of M*.
To be more concrete, this means that we wish to develop a
suitable function J on the set of closed terms such that
for all closed terms s,t, we have
s ≡ t if and only if J(s) = J(t).
For any closed term s we write s# for the result of
replacing all new constants in s by the common marker #.
Let s be a term. Let a be the least t# such that s ≡ t,
where we use some standard indexing of all finite sequences
of the relevant symbols.
Let V be the set of all closed terms t such that s# = t#.
Among t Œ V, s ≡ t is determined solely by the order type
of the subscripts of the new constants that appears in the
expression s = t. Therefore the set b(s) of all sequences of
subscripts of new constants in t Œ V with s ≡ t constitutes
a definable set in (D,<) with parameters. 77
By Lemma 3.12, we know that there is a unique set g(s) Õ D
which serves as parameters for a definition in (D,<) of
b(s), such that no proper subset so serves.
We then take J(s) to be the definition of b(s) in (D,<)
which uses exactly the parameters g(s) and which is least in
some standard indexing of definitions.
If s ≡ s' then it is easy to see that a(s') = a(s), b(s') =
b(s), g(s') = g(s), and J(s') = J(s).
Note that the values of J are finite sequences from D ». w.
Also note that the new constants lie in separate
equivalence classes; i.e, J(cx) = J(cy) ´ x = y. Hence the
values of J include D. So we can construe the resulting
model as having domain D, as required. QED
PHILOSOPY 532
PHILOSOPHICAL PROBLEMS IN LOGIC
LECTURE 8
11/20/02
Last time we gave some background about general domains,
and showed that the following conditions on a domain D
determine which sets of sentences in PC(=) are satisfiable
with domain D (i.e., Dsatisfiable):
D has at least two elements, has a linear ordering,
and has a pairing function.
In fact, the sets of sentences of PC(=) that are Dsatisfiable are exactly those that are consistent with INF.
1. Basic theory of predication.
We now want to get clear about the basic metatheory that we
will be using in order to discuss predication on the
universal domain W. We first present the theory BTP = basic
theory of predicates.
BTP is a two sorted theory based on objects and unary
predicates on the objects. The vocabulary of BTP consists
of
1) object variables xn, n ≥ 1;
2) predicate variables Pn, n ≥ 1, ranging over unary
predicates on objects; 78
3) constant object 0;
4) binary function symbol < > from objects to objects;
5) equality = between objects.
6) the usual connectives and quantifiers, as well as commas
and parentheses.
The object terms of BTP are given by
7) every object variable is an object term;
8) 0 is an object term;
9) if s,t are object terms then <s,t> is an object term.
The atomic formulas of BTP are
10) s = t, where s,t are object terms;
11) P(s), where P is a predicate variable and s is an
object term.
The formulas of BTP are:
12) atomic formulas of BTP are formulas of BTP;
13) if j,y are formulas of BTP then (ÿj), (j Ÿ y), (j ⁄ y),
(j Æ y), (j ´ y) are formulas of BTP;
14) if j is a formula of BTP, x is an object variable and P
is a predicate variable, then ("x)(j), ($x)(j), ("P)(j),
($P)(j) are formulas of BTP.
We refer to the language of BTP as L(BTP).
The axioms and rules of BTP are
15) the usual axioms and rules for predicate calculus based
on the language of BTP;
16) Pairing. <x,y> = <z,w> ´ (x = z Ÿ y = w);
17) Zero. ÿ<x,y> = 0;
18) Strict Comprehension. ($P)("x)(P(x) ´ j), where j is a
formula in L(BTP) with at most the free variable x.
We think of the object variables as ranging over absolutely
everything. 0 can be taken to be the number zero, or if
that is considered vague/meaningless, take it to be your
favorite object, or even "the idea of the universal
domain". Take <x,y> to be the ordered pair of x,y, or if
you prefer, the idea of: x followed by y. Take the
predicates to be predicates that sensibly applied to
absolutely everything. BTP is based on the idea that there
is a concept of predication that is being used which can be
referred to when designating predicates. This is reflected 79
in the use of quantifiers over predicates in Pure
Comprehension. We use the word "Pure" to indicate that no
free variables other than x are allowed in j.
We shall see that BTP is already a rather strong system in
that it is mutually interpretable with the formal system Z2
of f.o.m. In particular, one can faithfully develop
arithmetic, and arithmetic mathematics. One can go quite a
bit further, in some reasonable senses.
2. Identity of indiscernibles, singleton extension
principle.
We now take up the issue of the status of the identity of
indiscernibles. This is the principle
IIS) ("x,y)(x = y ´ ("P)(P(x) Æ P(y)).
THEOREM 2.1. BTP neither proves nor refutes IIS.
Proof: Let D be the set of all closed terms in the language
0,1,2,< >. There is a unique automorphism of (D,0,< >) that
sends 1 to 2 and 2 to 1. Take the predicates to be all
subsets of D that are fixed under this automorphism. I.e.,
the forward image under this automorphism is itself. QED
In fact, this proof will work for a rather innocent looking
extension of BTP, which we write as BTP'. Here Strict
Comprehension is extended as follows:
Pure Comprehension. ($P)("x)(P(x) ´ j), where j is a
formula in L(BTP) in which P is not free and where all free
object variables in j are x.
THEOREM 2.2. BTP' neither proves nor refutes IIS.
Now that we have introduced BTP', we can ask whether it
follows from BTP.
THEOREM 2.3. BTP does not prove BTP'.
Proof: Let D be the closed terms in 0,1,2,3,< >. Use as
predicates the P Õ D such that for some 1 £ i £ 4, P is
fixed under the six automorphisms of D that fix i and
permute all the other three elements of {1,2,3,4}. This
family of predicates is fixed under all 24 automorphisms of
D in the appropriate sense. Hence this construction is a 80
model of BTP. If the model satisfies BTP', then {1,2}
exists. However, {1,2} does not have the property that for
some 1 £ i £ 4, {1,2} is fixed under the automorphisms of D
that fix i and permute all of the other elements of
{1,2,3,4}. QED
The obvious axiom to add to BTP that will allow us to prove
IIS is the following principle of Singleton Extensions:
SEP) ("x)($P)("y)(P(y) ´ y = x).
THEOREM 2.4. BTP + SEP proves IIS.
Proof: Let x,y be such that ("P)(P(x) Æ P(y)). By SEP let P
be such that ("z)(P(z) ´ z = x). Then P(x). Hence P(y), and
so y = x. QED
Is BTP' + SEP consistent? Of course. Let D be the set of
all closed terms in 0,< >, and take the predicates to be
all subsets of D.
But does SEP follow from IIS? No, not even over BTP'.
THEOREM 2.5. BTP' + IIS does not prove SEP.
Proof: Probably this can be done without heavy duty
machinery, but here goes. Let M be a countable transitive
model of ZFC + V = L. Let M[x] be a generic extension of M
by adding a Cohen subset x Õ w. Now let D be S(w) in the
sense of M[x] and the predicates on D be the ordinal
definable subsets of S(w) in the sense of M[x]. Let 0 be ∅
and let < > be defined in the obvious way use a standard
pairing function on w. Then this defines a model of BTP'
since any set given by pure comprehension must be an
ordinal definable subset of S(w) in the sense of M[x].
However SEP is clearly false in this model since {x} is not
ordinal definable in the sense of M[x]. However IIS holds
since for any x,y Œ D, there exists n such that n Œ x ´ n
œ y. Let n Œ x\y. Then x Œ {z: n Œ z} and y œ {y: n Œ y}.
The other case is handled symmetrically. QED
THEOREM 2.6. BTP + SEP does not prove BTP'.
Proof: Again, probably this can be done without heavy duty
machinery, but here goes. We start with a countable
transitive model M of ZFC + V = L. We introduce in the
usual way, the generic set {{x11,x12,...},{x21,x22,...},...}, 81
where the x's are mutually Cohen generic subsets of w. We
take D to be S(w) in the sense of M' =
M[{{x11,x12,...},{x21,x22,...},...}] = M[E], with 0,< >
interpreted as before. The predicates are those subsets of
D such that for some u Œ E and x Õ w, the subset is ordinal
definable from E,u,x. Then BTP + SEP can be verified to
hold in this model. However, {x11,x12,...,x21,x22,...} does
not lie in this model, and so BTP' fails. QED
There is a very natural interpretation of BTP + SEP in BTP.
This is also an interpretation of BTP' + SEP in BTP'. This
is called the cross sectional interpretation.
Under this interpretation, the object sort remains fixed,
but the predicates are enlarged. The predicates are now
pairs P,x, where P is a predicate and x is an object. We
define (P,x)(y) if and only if P(<x,y>).
THEOREM 2.7. Under the cross sectional interpretation, BTP
+ SEP is interpreted in BTP. Also BTP' + SEP is interpreted
in BTP'.
Proof: We first prove the cross sectional interpretation
SEP in BTP. Fix x. We need to find P,y such that for all
(P,y)(z) ´ z = x. We will use y = x. So we need to find
such that for all z, P(<x,z>) ´ z = x. Take P such that
all w, P(w) ´ ($u)(w = <u,u>). This is provided by BTP. of
z,
P
for We next verify the cross sectional interpretation of BTP in
BTP. We must verify ($P)("x)(P(x) ´ j), where j has at most
the free variable x.
Fix j with at most the free variable x. Let the cross
sectional interpretation of ($P)("x)(P(x) ´ j) be
($P,y)("x)(P(<y,x> ´ j'), where j' has at most the free
variables x,y. Choose P such that ("z)(P(z) ´ ($x)(z =
<0,x> Ÿ j')). This P is provided by BTP.
For the second claim, we need to verify the cross sectional
interpretation of BTP' in BTP'. Let the interpretation of
($P)("x)(P(x) ´ j) be ($P,y)("x)(P(<y,x>) ´ j'), where j'
has at most the free object variables x,y, and may have
various free predicate variables other than P.
It clarifies matters to start with the sentence
("R1,...,Rk)($P)("x)(P(x) ´ j) 82
whose interpretation is
("R1,x1)...("Rk,xk)($P,y)("x)(P(<y,x>) ´ j')
where j' has at most the free variables R1,...,Rk,x,x1,...,
xk.
Let
j'' = ($x1,...,xk)(y = <x1,...,xk> Ÿ j').
Using BTP', fix P such that
("x,y)(P(<y,x>) ´ j'').
Now let R1,...,Rk,x1,...,xk be given. Let y = <x1,...,xk>.
Then
("x,y)(P(<y,x>) ´ ($x1,...,xk)(y = <x1,...,xk> Ÿ j')
and so
("x,y)(P(<y,x>) ´ j')
as required. QED
We now consider the strongest form of comprehension.
General Comprehension. ($P)("x)(P(x) ´ j), where j is a
formula of L(BTP) in which P is not free.
THEOREM 2.8. BTP' + SEP is equivalent to BTP with General
Comprehension.
Proof: The free object quantifiers in j can be replaced by
predicate variables because of SEP. QED
3. Pure and general predication. BTPpg.
The upshot of the previous section is the emergence of two
kinds of comprehension, one weak and one strong. The
difference is what kinds of free variables are allowed. In
the weakest, no free variables other than the comprehending
object variable is allowed. In the strongest, any free
variables are allowed except the beginning existential
predicate variable. 83
The proper way of sorting this out is to distinguish
between two kinds of predication.
We call the first, pure predication. Here we think of
predication in some sort of language, where we are not
entitled to simply refer to any object as we create,
discover, or contemplate, a pure predicate. If we do want
to refer to an object, we must define it: i.e., pick it out
uniquely with a predicate, and this predicate is also
subject to the same constraints.
We call the second, general predication. This is the one
that is most directly relevant to set theory and
mathematics. Here we can freely use any objects as
parameters. Infinitely many objects can also be used as
parameters provided that they together form an object.
However, if they, together, form an object, then we don't
need to consider infinitely many objects for this purpose.
Of course, this discussion is a bit murky, but it is
readily backed up by very satisfying formalisms.
The pure predication concept is formalized by our previous
BTP' with its Pure Comprehension. The general predication
concept is formalized by General Comprehension. We are not
asserting any kind of complete formalizations. Just that
these formalisms represent the principal points and
distinctions.
Both of these notions seem fundamental. It can be argued,
however, that pure predication is most fundamental, and
general predication is a derived notion. How derived? By
cross sections. I.e., a general predicate P is always given
by a pure predicate R together with an object x, where
("y)(P(y) ´ R(<x,y>).
At the other extreme might be a view that only general
predication, steeped in great history from f.o.m., set
theory, and mathematics, is coherent, and pure predication
is not.
We have decided to incorporate both notions into a single
theory, BTPpg. Here p is for pure and g is for general.
A number of questions arise when we consider BTPpg. Among 84
them is the question of the relationship between the pure
predicates and the general predicates. E.g., is it
necessarily the case that every general predicate is a
cross section of a pure predicate?
BTPpg is a three sorted theory based on objects, pure
(unary) predicates on objects, and general (unary)
predicates on objects. The vocabulary of BTPpg consists of
1) object variables xn, n ≥ 1;
2) pure predicate variables Ppn, n ≥ 1, ranging over pure
predicates on objects;
3) general predicate variables Pgn, n ≥ 1, ranging over
general predicates on objects;
4) constant object 0;
5) binary function symbol < > from objects to objects;
6) equality = between objects.
7) the usual connectives and quantifiers, as well as commas
and parentheses.
The object terms of BTPpg are given by
8) every object variable is an object term;
9) 0 is an object term;
10) if s,t are object terms then <s,t> is an object term.
The atomic formulas of BTPpg are
11) s = t, where s,t are object terms;
12) P(s), where P is a pure or general predicate variable,
and s is an object term.
The formulas of BTPpg are given by
13) atomic formulas of BTPpg are formulas of BTPpg;
14) if j,y are formulas of BTPpg then (ÿj), (j Ÿ y), (j ⁄
y), (j Æ y), (j ´ y) are formulas of BTPpg;
15) if j is a formula of BTPpg and x is an object variable
and P is a pure or general predicate variable, then
("x)(j), ($x)(j), ("P)(j), ($P)(j) are formulas of BTPpg.
We refer to the language of BTPpg as L(BTPpg).
The axioms and rules of BTPpg are
16) the usual axioms and rules for predicate calculus based
on the language of BTPpg; 85
17) Pairing. <x,y> = <z,w> ´ (x = z Ÿ y = w);
18) Zero. ÿ<x,y> = 0;
19) Pure Comprehension. ($Pp)("x)(Pp(x) ´ j), where j is a
formula in L(BTPpg) all of whose free variables are either
pure predicate variables other than Pp or x;
20) General Comprehension. ($Pg)("x)(Pg(x) ´ j), where j is
a formula in L(BTPpg) in which Pg is not free.
We also have the two important fragments BTPp and BTPg. The
first is the fragment in which no general predicate
variables are allowed. The second is the fragment in which
no pure predicate variables are allowed.
THEOREM 3.1. Every theorem of BTPpg without general
predicate variables is a theorem of BTPp. Every theorem of
BTPpg without pure predicate variables is a theorem of
BTPg.
Proof: We have essentially proved this already. Suppose we
have a model of BTPg. Then we can extend it to a model of
BTPpg without changing the objects and general predicates
by making the pure predicates the same as the general
predicates. Alternatively, suppose we have a model of BTPp.
Then we can extend it to a model of BTPpg without changing
the objects and pure predicates by making the general
predicates the cross sections of the pure predicates. QED
We define SEP and IIS for BTPpg in the same way as we did
before, of course referring only to pure predicates. For
general predicates, SEP and IIS are trivially true.
THEOREM 3.2. BTPpg proves SEP Æ IIS, and does not prove IIS
Æ SEP, and does not prove IIS.
Proof: Immediate from Lemmas 2.2, 2.4, 2.5, and Theorem
3.1. QED
Note that BTPpg proves that every pure predicate has the
same extension as some general predicate, using General
Comprehension.
4. Mathematical development of BTPpg.
We can develop a pure predicate of natural numbers in BTPpg
as follows. Consider the following condition on general
predicates: 86
Pg(0) Ÿ ("x)(Pg(x) Æ Pg(<x,0>)).
We let wp be the intersection of all general predicates
satisfying this condition. The general predicate that holds
of everything satisfies this condition.
We claim that the induction principle holds:
(Pg(0) Ÿ ("x)(Pg(x) Æ Pg(<x,0>))) Æ (wp(x) Æ Pg(x)).
To see this, let Pg obey the left side. By construction, wp
is contained in Pg.
As a consequence, we see that we have induction on wp with
respect to all formulas in L(BTPpg) since we have full
comprehension for general predicates in BTPpg.
Similar ideas allow us to develop definition by recursion
on wp. We have recursion resulting in general predicates,
where the recursion data is in terms of general predicates.
Alternatively, we have recursion resulting in pure
predicates, where the recursion data is in terms of pure
predicates.
In particular, we have wp,0,1,+,•,<,= as pure predicates,
and so arithmetic is developed "purely". In particular, we
have the successor axioms.
We can also develop finite sequences of objects purely.
Consider the following condition on general predicates:
Pg(0) Ÿ ("x,y)(Pg(0) Ÿ (Pg(x) Æ Pg(<x,y>))).
The idea is that 0 counts as the empty sequence.
We let FSp be the intersection of all Pg satisfying the
above condition. This serves as the definition of finite
sequence.
We can develop the length function from the extension of
FSp to the extension of wp. This will be a pure function.
We can go on to develop the satisfaction relation for any
structure whose domain is W. If the structure is a pure
structure, then the satisfaction relation will be pure. If
the structure is a general structure, then the satisfaction
relation will be general. The underlying syntax used for 87
this purpose is of course pure.
An important theorem of BTPpg is the Schroeder Bernstein
theorem. We take it in the following forms:
THEOREM 4.1. The following is provable in BTPpg. Let f:A Æ
B and g:B Æ A be pure oneone functions, where A,B are pure
sets (predicates). Then there is a pure oneone onto
function h:A Æ B. The same holds with "pure" replaced by
"general".
The idea here is to imitate the usual proof of the
Schroeder Bernstein theorem with its analysis of the orbits
of points.
5. Pure principle of symmetric arguments.
We now introduce the pure principle of symmetric arguments
to BTPpg.
PPSA. ("Pp)($x1 ≠ ... ≠ xk)(the conjunction over all 1 £ i1 ≠
... ≠ ik £ k of Pp(<x1,...,xk>) ´ Pp(<xi1,...,xik>), where k ≥
1.
We break this up with two parameters. Let k,r ≥ 1.
PPSA(k,r). ("Pp)($x1 ≠ ... ≠ xr)(the conjunction over all 1 £
i1 ≠ ... ≠ ik £ r of Pp(<x1,...,xk>) ´ Pp(<xi1,...,xik>).
In particular, PPSA(2,2) asserts
("Pp)($x ≠ y)(P(<x,y>) ´ P(<y,x>)).
We think of PPSA(k,r) as asserting that any pure predicate
of k variables is symmetric on r distinct arguments.
THEOREM 4.1. For all r ≥ 1, BTPp proves PPSA(1,r).
THEOREM 4.2. BTPp does not prove PPSA(2,2).
Proof: To see that BTPp does not prove PPSA(2,2), take D to
be the closed terms in 0,< >, and the predicates to be all
subsets of D. It suffices to see that in this model, there
is a linear ordering of D. QED
THEOREM 4.3. BTPp + PPSA is consistent. 88
Proof: Let D be the set of closed terms in 0,< > and
constants c1,c2,... . Let the predicates be all subsets of D
that are fixed under all automorphisms of D. These
automorphisms are given by permutations of the c's.
Obviously PPSA holds in this model. QED
Note that we can state PPSA(k,r) with parameters k,r, in
BTPp.
THEOREM 4.4. The following is provable in BTPp. If k £ k'
and r £ r' then PPSA(k',r') Æ PPSA(k,r).
PHILOSOPHY 532
PHILOSOPHICAL PROBLEMS IN LOGIC
LECTURE 9
11/27/02
We continue with a discussion of the principle of symmetric
arguments.
1. Pure principles of symmetric arguments.
Recall the pure principle of symmetric arguments:
PPSA. ("Pp)($x1 ≠ ... ≠ xk)(the conjunction over all 1 £ i1 ≠
... ≠ ik £ k of Pp(<x1,...,xk>) ´ Pp(<xi1,...,xik>), where k ≥
1.
PPSA(k,r). ("Pp)($x1 ≠ ... ≠ xr)(the conjunction over all 1 £
i1 ≠ ... ≠ ik £ r of Pp(<x1,...,xk>) ´ Pp(<xi1,...,xik>).
In particular, PPSA(2,2) asserts
("Pp)($x ≠ y)(Pp(<x,y>) ´ Pp(<y,x>)).
We think of PPSA(k,r) as asserting that any pure predicate
of k variables is symmetric on r distinct arguments.
Recall the following from last time:
THEOREM 1.1. For all r ≥ 1, BTPp proves PPSA(1,r).
THEOREM 1.2. BTPp does not prove PPSA(2,2).
THEOREM 1.3. BTPp + PPSA is consistent.
THEOREM 1.4. The following is provable in BTPp. If k £ k' 89
and r £ r' then PPSA(k',r') Æ PPSA(k,r).
We also consider the multiple forms of PPSA.
MPPSA(k,r). ("P1p),...,Pnp)($x1 ≠ ... ≠ xr)(the conjunction
over all 1 £ i1 ≠ ... ≠ ik £ r and 1 £ j £ n of
Pjp(<x1,...,xk) ´ Pjp(<xi1,...,xik>), where k ≥ 1.
THEOREM 1.5. MPPSA(k,r) follows from PPSA(k,r).
Proof: Let P1,...,Pn be given. We want to get r
indiscernibles for P1,...,Pn just using PPSA(k,r).
Let P(x1,...,xk) if and only if there exists 1 £ i £ n such
that x1,...,xk are not indiscernibles for Pi, and for the
least such i, Pi(x1,...,xk) holds.
Let x1,...,xk be indiscernibles for P. We assume that
x1,...,xk are not indiscernibles for P1,...,Pn. I.e., there
exists i such that x1,...,xk are not indiscernibles for
P1,...,Pn.
case 1. P(x1,...,xn). Let i be least such that x1,...,xk are
not indiscernibles for Pi. Then P(x1,...,xn). Let y1,...,yn
be any permutation of x1,...,xn. Then i is also least such
that y1,....,yk are not indiscernibles for Pi. Since
P(y1,...,yn), we have Pi(y1,...,yn). This establishes that
x1,...,xn are indiscernibles for Pi. This is a
contradiction.
case 2. ÿP(x1,...,xn). We know that i exists, and so
ÿPi(x1,...,xn). Also for any permutation y1,...,yn of
x1,...,xn, by the same argument we have ÿPi(y1,...,yn), since
ÿP(y1,...,yn). This is also a contradiction. QED
We now consider the logical relationships between the
PPSA(k,r).
THEOREM 1.6. Let k ≥ 2. PPSA(k,k) does not imply PPSA(2,k+1)
over BTPp.
Proof: The domain will be the closed terms in 0,< >,c1,...,
ck. The pure predicates will be those subsets of the domain
which are fixed under all automorphisms. The automorphisms
are given by permutations of c1,...,ck. We consider the
following pure binary relations. Let s,t be terms in D. The
syntax of s is defined to be the term with the subscripts 90
removed from all c’s. We assume an indexing of “syntaxes”
by natural numbers.
Let s be given. #(s) is the number of occurrences of c’s in
s. A position is given a number 1 £ i £ #(s). The
occurrences of c’s in s occupy positions 1,2,...,#(s).
A first position in s is a position whose constant differs
from the constant at all earlier positions.
Let a(s) be the set of all first positions in s. Note that
a(s) has at most k elements since there are only k
constants c1,...,ck.
We assume an indexing of finite sets of natural numbers by
natural numbers.
With these preliminaries, we are prepared to define some
pure binary relations.
P(s,t) iff the index of the syntax of s is < the index of
the syntax of t.
Q(s,t) iff the index of a(s) is < the index of a(t).
R(s,t) iff
i) s ≠ t;
ii) s,t have the same syntax;
iii) let i be least such that the c at position i in s
differs from the c at position i in t. Let cp be at position
i in s and cq be at position i in t;
iv) the first position where cp occurs in s is less than any
position where cq occurs in t.
Let 1 £ i £ k. Si(s,t) iff the constant at position equaled
to the ith element of a(s) and the constant at position
equaled to the ith element of a(t) both exist and are
unequal.
Now let t1 ≠ ... ≠ tk+1 be indiscernibles for P,Q,R,S1,...,Sk.
Using P, we see that the ti all have the same syntax. In
particular, the sequence of c’s appearing in the ti all have
the same length.
Using Q, we see that the a(ti) are all the same. 91
Suppose R(t1,t2). Then R(t2,t1). Let i be least such that the
c at position i in s differs from the c at position i in t.
Let cp be at position i in s and cq be at position i in t.
Then the first position where cp occurs in s is < any
position where cq occurs in t. Also the first position where
cq occurs in t is < any position cp occurs in t. This is a
contradiction.
Hence not R(t1,t2), not R(t2,t1). Therefore the first
position where cp occurs in s is ≥ some position where cq
occurs in t, and the first position where cq occurs in t is
≥ some position where cp occurs in s. Hence the first
position where cp occurs in s is the same as the first
position where cq occurs in t.
In particular, there exists i such that the constant at
position equaled to the ith element of a(t1) and the
constant at position equaled to the ith element of a(t2)
both exist and are unequal.
We now have Si(t1,t2), and there we have for all 1 £ b ≠ c £
k+1, Si(tb,tc). Hence the various constants that sit on the
position equaled to the ith element of the a(t)’s are all
different. This is a contradiction since there are k c’s
and k+1 t’s. So these k+1 indiscernibles cannot exist. QED
THEOREM 1.7. Let k ≥ 2 and n ≥ k. PPSA(k,n) does not imply
PPSA(k+1,k+1) over BTPp.
Proof: Let the domain D be the set of all closed terms in
0,< >, c1,c2,... . Construct a group G of permutations of
{c1,c2,...} such that
i)
ii) every permutation of any k element subset of
{c1,c2,...} can be extended to an element of G;
if f Œ G maps E into E, where E is a k+1 element
subset of {c1,c2,...}, then f is the identity on E. Now let the pure predicates be the subsets of D that are
fixed under all elements of G. Then all k element sequences
of distinct c’s look alike.
Some work is needed to finish this proof by showing that
PPSA(k+1,k+1) fails in this model. QED
Note that we have sort of encountered a yet stronger set of 92
principles that makes sense in the pure case. Namely, what
we call absolute indiscernibles.
UI(k,r). ($x1 ≠ ... ≠ xr)("Pp) (the conjunction over all 1 £
i1 ≠ ... ≠ ik £ k of Pp(<x1,...,xk>) ´ Pp(<xi1,...,xik>),
where k,r ≥ 1.
In Theorem 1.5, the model we construct obviously has
UI(k,k), but not PPSA(2,k+1). In the partial proof of
Theorem 1.6, we also obviously have UI(k,n) but presumably
not PPSA(k+1,k+1).
THEOREM 1.8. Let k ≥ 2. PPSA(k,k) does not imply UI(2,2)
over BTPp.
Proof: Let D be the set of all closed terms in 0,<
>,c1,c2,... . Let the pure predicates be the sets such that
for some n, that set is fixed under all automorphisms of D
that fix c1,...,cn. Then obviously MPPSA(k,k). However,
UI(k,k) fails. To see this, obviously there is a pure
predicate that holds of any given object and fails of any
other given object. QED
2. Pure linear orderings.
Clearly PPSA(2,2) alone is enough to refute the existence
of a linear ordering of W, over BTPp.
THEOREM 2.1. The nonexistence of a linear ordering is not
sufficient to prove PPSA(2,2) over BTPp.
Proof: Let D be the set of all closed terms in 0,<
>,c1,c2,... . Construct a “generic” or “random” binary
relation on the c’s such that for all i ≠ j, R(ci,cj) ´
ÿR(cj,ci). This relation has plenty of automorphisms. Let G
be the group of all of its automorphisms. Let the pure
predicates be the sets that are fixed under all elements of
G. To see that there is no linear ordering in this model,
it suffices to show that there is no linear ordering on
just the c’s that is fixed under all of the automorphisms
in G. But there is an automorphism in G that fixes any
finite set of c’s and moves any given remaining c into any
other given remaining c. The automorphisms of a linear
ordering cannot be that flexible. To see that PPSA(2,2)
fails, consider the following relation R on terms s,t.
P(s,t) if and only if s,t have the same syntax and at the 93
first place they differ and R holds at the constant in s at
that position comma the constant in t at that position, or
the syntax of s is less than the syntax of t.
Let s,t be indiscernibles for P.
case 1. s,t have the same syntax. Then clearly P(s,t) ´
ÿP(t,s), since the second disjunct does not have any
effect.
case 2. s,t do not have the same syntax. Then clearly
P(s,t) ´ ÿP(t,s), since the first disjunct does not have
any effect.
In either case, we have a contradiction. QED
3. General principles of symmetric arguments; general linear
orderings.
GPSA. ("Pg)($x1 ≠ ... ≠ xk)(the conjunction over all 1 £ i1 ≠
... ≠ ik £ k of Pg(<x1,...,xk>) ´ Pg(<xi1,...,xik>), where k ≥
1.
GPSA(k,r). ("Pg)($x1 ≠ ... ≠ xr)(the conjunction over all 1 £
i1 ≠ ... ≠ ik £ r of Pg(<x1,...,xk>) ´ Pg(<xi1,...,xik>).
In most cases, we never used anything that distinguishes
the pure from the general case, and so all of the results
go through virtually unmodified. The exception is the
discussion of absolute indiscernibles. These are outright
inconsistent if we use general predication.
THEOREM 3.1. GPSA does not follow from PPSA over BTPpg. In
fact, it is consistent to assume each UI(k,k) and there is
a general linear ordering, over BTPpg.
Proof: Let D be the set of all closed terms in 0,< >,{cp: p
Œ Q}. Let the pure predicates be the sets that are fixed
under all automorphisms. Let the general predicates be the
sets such that for some finite subset of the cp’s, are fixed
under all increasing order isomorphisms of Q into Q that
fix the elements of that finite subset. The usual linear
ordering of the c’s will be a general predicate. However,
the cp’s will form universal indiscernibles with respect to
pure predicates. One has to verify BTPpg. QED
We say that a general (pure) predicate P is finite iff 94
there is a finite sequence x (an object) such that
("y)(P(y) Æ y is a
term in x).
We say that a general (pure) predicate P is infinite iff it
is not finite.
THEOREM 3.2. BTPpg proves that a general predicate is
finite iff its extension is in general oneone
correspondence with a proper initial segment of N. BTPpg
does not prove all finite pure predicates are in pure oneone correspondence with a proper initial segment of N.
Proof: From a general oneone correspondence with a proper
initial segment of N, one can get the required finite
sequence by induction. And one can get the general oneone
correspondence as a predicate of ordered pairs directly. In
the pure case, let D be the set of closed terms in 0,<
>,c1,c2. Let the pure predicates be the subsets of D that
are fixed under both automorphisms. Let the general
predicates be all subsets of D. Then {c1,c2} is pure but no
twotuple with both of them can be pure. QED
A minimally infinite general predicate is an infinite
general predicate P such that no general predicate splits
P. I.e., for any general predicate Q, either
i)
ii) there is a finite sequence x such that ("y)((P(y) Ÿ
Q(y)) y is a term in x); or
there is a finite sequence x such that ("y)((P(y) Ÿ
ÿQ(y)) y is a term in x). THEOREM 3.3. BTPg proves: if $ minimally infinite general
predicate then GPSA. Converse not provable in BTPg.
Proof: The first claim is obvious. For the second claim,
let D be the closed terms in 0,< >, c1,c2,... . Let the
general predicates be the subsets of D that are fixed under
all automorphisms that, for some i, fix all but the
multiples of 2i. Then GPSA holds, but there is no minimally
infinite general predicate. QED
THEOREM 3.4. BTPg does not prove or refute $ a minimally
infinite general predicate. 95
Proof: Let D be as usual. We can use all subsets of D, in
which case it is false. Or we can use all subsets of D such
that for some finite set of c’s, it is fixed under all
automorphisms that fix each element of the finite set. Then
the set of all c’s is a minimally infinite general
predicate in this model. QED
An absolute POI (predicate of indisernibles) is a pure
predicate P where any pure predicate holds or fails of any
two equal length finite sequences of distinct objects from
the extension of P.
THEOREM 3.5. BTPp does not prove or refute the existence of
an infinite absolute POI.
Proof: Let D be the set of closed terms in 0,< >, c1,c2,...
. If we take the pure sets to be all subsets of D, then we
don’t have an infinite absolute POI. On the other hand, if
we take the pure predicates to be the sets fixed under all
automorphisms, then {c1,c2,...} is a POI.
4. Satisfiability of universal sentences.
We now get back to logic. Recall the last substantial
theorem about our main topic that we proved. In current
terminology it is this.
We use Dp satisfiability for satisfiability in domain D
with pure relations and functions. We use Dg satisfiability
for satisfiability in domain D with general relations and
functions. Recall that W is the universal domain.
THEOREM 4.1. BTPpg proves that the following are equivalent
for any pure domain D.
i)
the sets of sentences of PC(=) that are Dp satisfiable
are exactly the sets of sentences of PC(=) that are Np
satisfiable;
ii) the sets of sentences of PC(=) that are Dp satisfiable
are exactly the sets of sentences of PC(=) that are
consistent with INF(=);
iii) D has at least two elements, has a pure linear
ordering, and has a pure pairing function.
The same result holds for general domains D and with “pure”
replaced by “general” and with Dp replaced by Dg.
The extra trouble we went to in order to avoid factoring
out by a congruence relation at the end of the proof allows 96
us to conclude that the proof can be given in BTPpg.
By Wp satisfiable, we mean that we have a model with domain
W and pure relations and functions. By Wg satisfiable, we
mean that we have a model with domain W and general
relations and functions.
SYM(=) consists of the following set of axioms. Let k ≥ 1
and j be a formula in PC(=) with free variables among
x1,...,xk. We have the axiom
($x1 ≠ ... ≠ xk)(conjunction of (j(x1,...,xk)
´j(xp1,...,xpk))),
where the conjunction ranges over all permutations p of
1,...,k.
We also consider the subsystem QFSYM(=) which is the same
as SYM(=) except that j is required to have no quantifiers.
In addition, we also consider the multiple forms of these
two systems, MSYM(=) and MQFSYM(=). These are formulated in
the obvious way using finitely many formulas (quantifier
free formulas) j at once.
LEMMA 4.2. SYM(=) and MSYM(=) are logically equivalent.
QFSYM(=) and MQFSYM(=) are logically equivalent.
Proof: By adapting the proof of the equivalence of PPSA and
MPPSA. QED
THEOREM 4.3. BTPp proves that every set of universal
sentences of PC(=) consistent with QFSYM(=) is Wp
satisfiable.
Proof: Let S be a set of universal sentences of PC(=)
consistent with ATSYM(=). Let T be the following theory in
PC(=) with new constant symbols c1,c2,... . The axioms of T
are the sentences
j(d1,...,dk) ´ j(e1,...,ek),
where k ≥ 1, d1,...,dk are distinct elements of {c1,c2,...},
e1,...,ek are distinct elements of {c1,c2,...}, and j is a
quantifier free formula in PC(=) with the k free variables
x1,...,xk. 97
We claim that S + T is consistent in PC(=) extended to
accommodate the expanded language. To see this, it suffices
to prove that every finite subset of S + T is consistent.
Note that this follows immediately from the consistency of
MQFSYM(=) + S by existentially quantifying out the relevant
constants. The consistency of MQFSYM(=) + S is from Lemma
4.2.
Now let M be a model of S + T. Since the axioms of S + T
are universal, we can assume that the domain D of M is
generated from the new constants c1,c2,... by the constant
and function symbols of PC(=).
Working in BTPp, we now construct a model M* whose domain
consists of the closed terms generated by the constants and
functions of PC(=) and the additional constants c*x, for
every object x. The interpretation of the constant and
function symbols of PC(=) are obvious, and the
interpretation of the relation symbols of PC(=) are given
by reference to M. The relevant additional constants are
mapped oneone back into the new constants c1,c2,... and the
truth value that results is imitated.
As in the proof of Theorem 4.1, we thus obtain a model
whose equality relation is not identity. We avoid having to
factor out by finding a canoncial “name” for each
equivalence class just as in the proof of Theorem 4.1. In
this case, we need only develop a canonical “name” for
every definable relation in the set of additional constants
c*x under equality. This is simpler, and follows from the
development of canonical names in connection with Theorem
4.1, where we did this under a dense linear ordering
without endpoints. As in Theorem 4.1, the proof is
completed by intensive use of the Shroeder Bernstein
theorem in BTPp. QED
Note that according to Theorem 4.3, we have a lower bound
on the sets of universal sentences of PC(=) that are Wp
satisfiable. If we accept PPSA, then our mission is
accomplished for sets of universal sentences of PC(=). (At
least under the pure interpretation).
THEOREM 4.4. The following are equivalent over BTPp.
i)
The sets of universal sentences of PC(=) that are Wp
satisfiable are exactly those that are consistent with
SYM(=) (QFSYM(=), MSYM(=), MQFSYM(=));
ii) Every universal sentence of PC(=) that is Wp 98
satisfiable is consistent with SYM(=);
iii) The pure principle of symmetric arguments, PPSA,
holds;
iv) The multiple pure principle of symmetric arguments,
MPPSA, holds.
Furthermore, we can replace “purely” and “pure” throughout
by “generally” and “general”.
In i), we can use any choice of the four systems shown.
Proof: This follows from Theorem 4.3 together with the
following observation. Under PPSA, every structure with the
universal domain satisfies SYM(=). Thus we see that BTPp
proves iii) implies i). To see that ii) implies iii) in
BTPp, suppose ii) holds. Suppose PPSA fails. Then an
instance of SYM(=) has a pure countermodel in the universal
domain. Hence an instance of SYM(=) is not consistent with
SYM(=). Hence SYM(=) is inconsistent. This is a
contradiction, for we can take as a model of SYM(=), the
natural numbers where the constants, relations, and
functions are defined in a trivial way; e.g., the constants
are all 0, the relations hold universally, and the
functions are all first projection functions. We have
already shown that iii) and iv) are provably equivalent in
BTPp. QED
5. Provable Wp satisfiability.
This section concerns the determination of a modified
notion of Wp satsifiability which does not require a
principle like PPSA.
Let T be any extension of BTPp in its language.
We call a sentence j of PC(=) provably Wp satisfiable over
T if and only if T proves
j is Wp satisfiable in the universal domain.
LEMMA 5.1. The following is provable in Peano arithmetic.
Let j be a universal sentence of PC(=). Then j is
consistent with SYM(=) if and only if j has a model with
exactly n elements other than the interpretation of
constants in j, where r is the relational type of j and n is
the highest arity of all relation symbols appearing in j,
and any permutation of the domain that fixes the
interpretation of the constants in r is an automorphism of 99
the model.
Proof: Suppose j is consistent with SYM(=). In PA, we can
build a model of SYM(=) + j (with complete diagram), and
cut down to the needed finite submodel with n elements
other than the interpretation of constants in j. Symbols
outside the relational type of j are interpreted trivially.
On the other hand, suppose j has a model as indicated. Then
the model can be stretched to provide an infinite set of
indiscernibles. Then SYM(r) is automatic. Again the symbols
outside r can be interpreted trivially.
THEOREM 5.2. Let j be a universal sentence of PC(=). Then j
is provably Wp satisfiable over BTPp if and only if j is
consistent with SYM(=).
Proof: Let j be as given. Suppose j is provably Wp
satisfiable over BTPp. If j is not consistent with SYM(=),
then j can be refuted from SYM(=), in which case BTPp +
PPSA proves that j is not Wp satisfiable. In particular,
BTPp + PPSA is inconsistent, which is a contradiction.
Suppose j is consistent with SYM(=). By Lemma 5.1, BTPp
proves that j is consistent with SYM(=). By Theorem 4.3, j
is provably Wp satisfiable over BTPp. QED
THEOREM 5.3. Let j be a universal sentence of PC(=). Then j
is provably Wp satisfiable over BTPp if and only if it is
provable in Z2 that j is consistent with SYM(=). Also j is
provably Wp satisfiable over BTPp plus the true sentences
of arithmetic if and only if j is consistent with SYM(=).
PHILOSOPHY 532
PHILOSOPHICAL PROBLEMS IN LOGIC
SUPPLEMENARY NOTES
11/8/02
11/12/02
As indicated in Phil 532 Lecture 6, we will discuss
subjective isomorphisms here. However, while developing
this theory of subjective isomorphisms, we came across yet
another approach that does not involve subjective concepts.
In fact, this most recent approach seems perhaps the most
promising of all. However, it may merge with the subjective
isomorphisms approach at a later time. 100
Nevertheless, we think that the subjective isomorphisms
idea has enough merit to warrant our discussing it in the
first two sections of this supplement.
Then we move on to a newer topic: mereological foundations
of set theory.
1. Subjective automorphisms in arithmetic.
Recall the system IS0 of bounded arithmetic. The language is
0,S,+,•,<,=. We have some basic axioms (e.g., Robinson's Q)
together with induction for all S0 or bounded formulas.
I.e., the quantifiers are all bounded to variables. (We
could bound the quantifiers to terms as well).
We expand the language of IS0 by adding a new sort for
subjective unary functions from N into N. We assume
standard axioms and rules of inference of logic for this
language.
We use the word "subjective" because we are definitely not
going to allow use of these functions in the induction
scheme. This would lead to an immediate inconsistency with
the other axioms we use.
We define "F is an automorphism" as
F(0) = 0 Ÿ ("x,y)(F(S(x)) = S(F(x)) Ÿ F(x+y) = F(x)+F(y) Ÿ
F(x•y) = F(x)•F(y) Ÿ (x < y ´ F(x) < F(y)) Ÿ
(x = y ´ F(x) = F(y))).
Obviously there are no automorphisms other than the
identity, by the obvious argument by induction. However,
there are, arguably, subjective ones; i.e., outside the
realm of objective thought. Or, putting it differently,
outside the realm of mathematical thought. Another way of
looking at this is that their are subjective automorphisms
with a shift in point of view. I.e., a change in the
reference frame. We are suggesting that their is a coherent
relativistic foundation for mathematics. What happens to
f.o.m. if we make moves analogous to the moves made in
physics by Einstein's special and general relativity
theories?
We now present the theory BA# (bounded arithmetic sharp).
1. Bounded arithmetic (i.e., IS0). 101
2. ("x)($y)($F)(F is an automorphism Ÿ ("z < x)(F(z) = y) Ÿ
("z > y)(F(z) ≠ z)).
THEOREM 1.1. BA# proves every axiom of PA (Peano
arithmetic). In fact, the theorems of BA# in the language
of BA are exactly the theorems of PA. BA# and PA are
equiconsistent.
Of course, one can read this result as a result in the
model theory of arithmetic as follows.
THEOREM 1.2. Every model of BA which has automorphisms
fixing any given bounded set, but whose fixed points are
bounded, is a model of PA. In fact, any consistent
extension of PA has a countable model with automorphisms
whose set of fixed points are arbitrarily high proper
initial segments.
For technical reasons, we will need EFA rather than BA. EFA
= IS0(exp), which is in the language 0,S,+,•,2x,<,=.
Let the language of EFA* be the language of EFA together
with the unary function symbols F,G. We will not need
variables over subjective functions. F,G are to represent
two specific subjective functions. The axioms of EFA* are:
1. EFA.
2. F,G are automorphisms (with respect to the language of
EFA).
3. F,G each fix exactly two different proper initial
segments.
Let EFA* be the resulting formal system.
THEOREM 1.3. Every finite fragment of PA is interpretable
in EFA*. EFA* and PA are equiconsistent. In fact, using
some standard tricks, PA is interpretable in EFA#. These
statements are provable in EFA.
Proof: We first give a construction of a model of PA as a
specific initial segment of any model M of EFA*. We then
indicate how to turn this proof into the desired
interpretation.
Let M have automorphisms f,g, and let the fixed points of f
be the proper initial segment I, and let the fixed points
of g be the proper initial segment J. Assume that I ⊇≠ J. 102
It is clear that I,J are each closed under exponentiation.
Let x Œ I\J and let y = 2^2^x. Then y Œ I\J.
Let E be the set of elements of dom(M) generated by f,g,f1,g1,y. Then every element of E is a double power of 2.
We claim that for all z Œ E, f(z) ≠ z or g(z) ≠ z. We prove
this by induction on z.
case 1. z = y. Then g(y) ≠ y since y œ J.
case 2. z = f(w), f(w) ≠ w.
case 3. z = f(w), g(w) ≠ w. If z Œ I then y Œ I. This is
because I is closed under f.
case 4. z = f1(w), f(w) ≠ w.
case 5. z = f1(w), g(w) ≠ w.
case 6. z = g(w), g(w) ≠ w.
case 7. z = g(w), f(w) ≠ w.
case 8 z = g1(w), g(w) ≠ w. case 9 z = g1(w), f(w) ≠ w. Let z Œ E. If f moves x then f(x) > x or f1(x) > x. If g
moves x then g(x) > x or g1(x) > x. Hence E has no
greatest element. Since every element of E is a double
power of 2, E defines an initial segment K which is closed
under 0,1,+,•,<. We now show that K satisfies PA.
We can view E as
of automorphisms
from f,g,d. Note
automorphisms of
0,S,+,•,<. the set of all values at y of the group G
of M generated by composition and inverse
that all such automorphisms are also
MK, where the structure MK uses only Let z Œ E. Then z = h(y) for some h Œ G. Now consider the
automorphism hfh1, which is also in G.
Note that hM is an isomorphism from (K,fK) onto (K,hfh1K). Since every element of [0,y] is a fixed point of fK,
we see that every element of [0,z] is a fixed point of hfh 103
1K. Also, since the fixed points of fK form a proper
initial segment of K, the fixed points of hfh1K form a
proper initial segment of K.
We have thus shown that K has the following property. There
are automorphisms of K whose fixed points are arbitrarily
long proper initial segments of K.
This is enough to ensure that K satisfies PA. To see this,
we use that K satisfies IS0 = bounded arithmetic. We will
not need exponentiation on K.
Suppose K satisfies ISk, k ≥ 0. In order to establish ISk+1
in K, it suffice to establish in K that
($b)("i £ x)(($y)(j(i,y)) Æ ($y £ b)(j(i,y)))
for any Pk formula j with parameters in K.
Let f be an automorphism of K that is the identity on
exactly the proper initial segment I of K, where x Œ I.
Suppose i £ x and ($y)(j(i,y)) holds in K. Then by the
induction hypothesis, let y be least such that j(i,y) holds
in K. Then y is definable in K from i, and so y is a fixed
point of f. Hence y Œ I.
We have thus established in K that
("i £ x)(($y)(j(i,y)) Æ ($y Œ I)(j(i,y)))
and so, letting I Õ [0,b], we have in K,
("i £ x)(($y)(j(i,y)) Æ ($y £ b)(j(i,y))). We have thus established that K satisfies PA.
We now indicate how to turn this proof into the desired
interpretation of any finite fragment T of PA into EFA*.
The specific place requiring modification is in the
construction of E and K. Not only do we have access to very
little induction (EFA), we have absolutely no access to any
kind of induction involving the automorphisms f,g and their
sets of fixed points I,J. .... QED
2. Subjective automorphisms in set theory. 104
We begin with BST = bounded set theory. We will be
minimalistic about things and use only Œ, and no
extensionality. The axioms are
1. Bounded separation. ($x)("y)(y Œ x ´ (y Œ a Ÿ j)),
where j is a bounded formula (i.e., all quantifiers are
bounded).
We now formulate BST#. The language will be that of BST
together with variables over subjective functions. We write
F is an automorphism for
("x,y)((x Œ y ´ F(x) Œ F(y)) Ÿ (x = y ´ F(x) = F(y))).
The axioms of BST# are
1. BST.
2. ("x)($y)($F)(("z Œ x)(F(z) = z) Ÿ ("z œ y)(F(z) ≠ z)).
THEOREM 2.1. ZFC is interpretable in BST#. BST# is
interpretable in NBG (von Neumann Bernays Gödel class
theory). ZFC and BST# are equiconsistent.
3. Mereological foundations.
We now shift attention to a new approach, independent at
the moment, of subjective isomorphisms.
In pure mereology, one considers only parts of the whole,
where the null part is banned. Thus all objects are
"parts". One has only the part/whole relation and no other
concepts. The part/whole relation is assumed to be
transitive and reflexive.
In mereology, one can take equality as primitive or
defined. If primitive, then one relies on the usual
classical first order predicate calculus with equality.
Otherwise it is defined as
x ≡ y ´ (x £ y Ÿ y £ x).
It is easily seen that the usual axioms for equality can be
derived for this interpretation of equality, so the two
approaches are in an appropriate sense equivalent. We
choose to take = as primitive.
There are two kinds of axiomatizations of pure mereology. 105
One is the axioms corresponding to a Boolean algebra
(without 0). The other is the least upper bound principle
for all formulas in the language. This scheme is called
fusion. The least upper bound is called the mereological
sum. The important classical theorem is that these two
axiomatizations are logically equivalent.
It is important to recall that there are some important
additional axioms that make the system complete. An atom is
defined to be a part which is a part of any of its
subparts. If we specify axiomatically an exact finite
number of atoms, and whether there exists a part with no
part that is an atom, then we get a complete system.
Alternatively, if we specify axiomatically that there are
infinitely many atoms (with infinitely many axioms), and
whether there exists a part with no part that is an atom,
then we also get a complete system. Furthermore, these are
the only complete extensions.
Now let us be more formal. Let M (pure mereology) be the
following system in the language L(£,=), which is the
ordinary classical first order predicate calculus with £ and
equality. We assume the usual axioms and rules for this
language.
M1. £ is reflexive, transitive, and has no minimum element.
M2. (x £ y Ÿ y £ x) Æ x = y.
M3. ($x)(("y)(j Æ y £ x) Ÿ ("z)(("y)(j Æ y £ z) Æ x £
z)), where j is a formula of L(£) in which x,z are not free.
We now introduce a naming system to mereology. All names
are to be atoms, but we do not require that every atom be a
name. A name can only name one part. (Everything is a
part).
Formally, we let MN be mereology with naming. The language
is L(£,NA,=), where NA is a binary relation. We assume the
standard axioms and rules of inference for classical
predicate calculus based on this language. The intended of
NA is NA(x,y) iff x is a name of y.
The axioms of MN are
1. £ is reflexive, transitive, and has no minimum element.
2. (x £ y Ÿ y £ x) Æ x = y.
3. ($x)(("y)(j Æ y £ x) Ÿ ("z)(("y)(j Æ y £ z) Æ x £ z)),
where j is a formula of L(£,NA) in which x,z are not free. 106
4. NA(x,y) Æ x is an atom.
5. (NA(x,y) Ÿ NA(x,z)) Æ y = z.
We say that y is named if and only if there exists x such
that NA(x,y). We say that x is an atomic part of y if and
only if x is an atom and x is a part of y.
We now sketch a proof in MN that something is not named.
LEMMA 3.1. The following is provable in M. Every atomic
part of a fusion is a part of the parts fused.
Proof: Let x be an atom that is part of a fusion, but not
part of the parts fused. Then all of the things fused are
part of the complement of x. Hence the fusion is a part of
the complement of x. This contradicts that x is part of the
fusion. QED
THEOREM 3.2. The following is provable in MN.
($x)("y)(ÿNA(y,x)).
Proof: Suppose every part has a name. Consider the fusion S
of all names which are not a part of the part that it
names. Let x be the name of S. First suppose that x is a
part of S. Since x is an atom, it must be a part of one of
the names which are not a part of the part that it names.
Hence x is not a part of the part that it names. Hence x is
not a part of S.
Finally suppose that x is not a part of S. Then x is not a
part of the part that it names. Hence x is a part of S.
Both suppositions have led to a contradiction. QED
Thus we see that the refutation of "every part has a name"
corresponds closely to Russell's and Tarski's paradoxes.
We now define the bounded formulas of L(£,NA,=) by the
following inductive clauses.
i) every atomic formula is bounded.
ii) if j,y are bounded then so are ÿj, j ⁄ y, j Ÿ y, j Æ
y, j ´ y.
iii) if j is bounded and y,z are distinct variables then ($y
£ z)(j) and ("y £ z)(j) are bounded.
We are now ready to give the axiom scheme of "local 107
completeness". This says that the naming system is "locally
complete".
6. Local completeness. j Æ ($z1 £ y1)...($zk £
yk)(j[y1,...,yk/z1,...,zk] Ÿ z1,...,zk are named), where j
is a bounded formula of L(£,NA,=), y1,...,yk is an
enumeration of the free variables of j, the 2k variables
y1,...,yk,z1,...,zk are distinct, and z1,...,zk do not
appear in j.
The resulting system 16 is MLCN (mereology with locally
complete naming).
THEOREM 3.3. MLCN corresponds roughly to a second order
indescribable cardinal. In particular, ZFC is interpretable
in MLCN. We also have equiconsistency. In the appropriate
sense, MLCN and ZFC plus roughly an indescribable cardinal
prove the same arithmetic sentences and more.
We are working on a very striking stronger principle to
replace 6 which would put us far beyond measurable
cardinals. ...
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.
 Fall '08
 JOSHUA
 Math, Logic

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