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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 536
PHILOSOPHY OF MATHEMATICS
LECTURE 1
9/25/02
1. LOGIC.
What are we trying to do with logic?
In this seminar, we concentrate on mathematical reasoning. It
appears that certain reasoning principles are used not only
in mathematics, but in a very wide variety of contexts
outside mathematics.
On the other hand, certain other principles of mathematical
reasoning seem peculiar to mathematics.
These are normally in the form of principles of set
existence.
This distinction between logic and mathematics is subject to
various criticisms and can be given various defenses.
Nevertheless, the division seems natural enough and is
commonly adopted in presentations of the standard foundations
for mathematics.
In particular, when presenting the crucial formal system ZFC
of Zermelo Frankel set theory with the axiom of choice, one
writes down 9 axioms (some are axiom schemes), and says "the
axioms and rules of first order predicate calculus with
equality and epsilon are understood".
What we try to do in logic (of the sort under consideration)
is to first identify a family of statement forms. In a
statement form, certain components are identified with
special names referring to "logical operations", and other
components are identified as atomic, with no internal
structure. 2
The idea is that the meaning of a statement depends,
uniformly, on the meaning of the atomic components and the
way in which the logical operations operate on meanings.
Two primary examples of "logics" stand out in f.o.m. so much
that in this seminar we will avoid the serious difficulties
involved in talking about "logics" in any substantial
generality.
These are propositional calculus and predicate calculus. The
first is subsumed by the second, but there are good reasons
to discuss it first.
Incidentally, sometimes the so called higher order predicate
calculus is considered rather than what we mean by (first
order) predicate calculus. For our purposes, this should be
reduced to (first order) predicate calculus together with a
dose of set theory. Because of the set theoretic aspect, it
is extremely ill behaved in comparison with (first order)
predicate calculus.
There is another brand of propositional calculus and
predicate calculus called intuitionistic propositional
calculus and intuitionistic predicate calculus. Current
mathematics is far from intuitionistic, although there is no
doubt that significant fragments of mathematics are
intuitionistic, and that it is quite interesting to see what
can or cannot be proved intuitionistically. This has
ramifications for the existence of algorithms associated with
mathematical theorems. We will only briefly touch on this
today, and it will play no (little) role later in the
seminar.
Since propositional calculus, PROP, is a primary example of a
logic, it has a syntactic side and a semantic side which must
not be confused.
On the syntactic side, we need to specify the vocabulary. The
symbols of PROP are taken to be
i) the "connectives" ÿ Ÿ ⁄ Æ ´;
ii) the "atoms" pn, where n ≥ 0.
These five connectives are read: not, and, or, if then, iff. 3
We have to describe what the statements look like. They are
defined inductively as follows.
1) any atom is a formula of PROP;
2) if A,B are formulas of PROP, then so are ÿ(A), (A Ÿ B), (A
⁄ B), (A Æ B), (A ´ B);
3) the only way to be a formula of PROP is through 1),2).
This kind of definition is completely typical of the grammar
of formal languages. Such grammars have been treated
systematically in great generality in the formal language
world (principally part of the computer science community).
There are two standard ways of making this rigorous, one from
"above", and one from "below".
From above, we define the formulas of PROP as the elements of
the least set of finite strings of symbols of PROP that
reflect 1) and 2). One proves that there is such a least set.
From below, we consider finite rooted trees whose terminal
vertices are labeled with atoms, and whose nonterminal
vertices are suitably labeled with connectives. The formulas
of PROP are those finite strings of symbols of PROP that
correspond to such "parse trees" in the obvious way.
A unique parsing theorem needs to be proved. In fact, there
is a well understood theory concerning conventions for the
reduction of parentheses needed in order to have unique
parsing  so called precedence grammar theory.
So far we have only discussed the syntactic side. Coming now
to the semantic side, we want to discuss what it means for a
given set of formulas of PROP to logically imply a given
formula of PROP.
This is defined in terms of truth assignments. A truth
assignment is a mapping from the set of atoms to the two
element set {T,F}.
We inductively define Sat(A,f), which means that A is
satisfied under the truth assignment f. Truth assignments are
thought of as interpretations of PROP.
The inductive definition proceeds as follows. 4
Sat(pn,f) iff f(pn) = T.
Sat((A,f) iff not Sat(A,f).
Sat(A Ÿ B,f) iff Sat(A,f) and Sat(B,f).
Sat(A ⁄ B,f) iff Sat(A,f) or Sat(B,f).
Sat(A Æ B,f) iff not Sat(A,f) or Sat(B,f).
Sat(A ´ B,f) iff (Sat(A,f) and Sat(B,f)) or (not Sat(A,f)
and not Sat(B,f)).
Let X be a set of formulas of PROP and A be a formula of
PROP. We write Sat(X,f) to indicate that Sat(B,f) holds for
all B Œ X. We say that X logically implies A iff for all f,
if Sat(X,f) then Sat(A,f).
We say that A is a tautology iff for all f, Sat(A,f).
There is a set of axioms and rules of inference which is
complete in the following sense.
WEAK COMPLETENESS. Every tautology is provable using these
axioms and rules.
WEAK SOUNDNESS. Every formula of PROP provable using these
axioms and rules is a tautology.
STRONG COMPLETENESS. If a given set of formulas of PROP
logically implies a given formula of PROP then that formula
can be derived from the set using these axioms and rules.
STRONG SOUNDNESS. If a given formula of PROP can be derived
from a given set of formulas of PROP using these axioms and
rules, then that set logically implies that formula.
Weak and strong completeness are equivalent because of a
purely semantic fact.
SEMANTIC FINITENESS. A given set of formulas of PROP
logically implies a given formula of PROP iff some finite
subset of the set does.
There are other logical relations between these standard
facts.
There are a number of such axioms and rules of inferences for
PROP in textbooks and elsewhere, none of which are all that
memorably neat. But here is a (new?) twist on this
axiomatization problem. 5 We will give a Hilbert style axiomatization. We use two
rules:
a. Modus ponens. From A and A Æ B derive B.
b. Substitution. From any axiom A derive every substitution
instance of A.
But what of the axioms?
c. All tautologies with at most 6 occurrences of atoms.
It would be interesting to see just how small a number we can
use instead of 6.
We now move on to predicate calculus. Here we use variables
xn, n ≥ 0, constant symbols cn, n ≥ 0, relation symbols Rnm of
arity n ≥ 1, where m ≥ 0, and function symbols Fnm of arity n
≥ 1, where m ≥ 0.
Terms are inductively defined using the constant and function
symbols as follows.
1. Each cn is a term.
2. If t1,…,tn are terms then Fnm(t1,…,tn) is a term.
Atomic formulas are of the form
Rnm(t1,…,tn).
In predicate calculus with equality, the atomic formulas are
of the forms
Rnm(t1,…,tn)
s=t
where s,t,t1,…,tn are terms.
The formulas are inductively defined as follows.
a. Every atomic formula is a formula.
b. If A,B are formulas then so are (ÿA), (A Ÿ B), (A ⁄ B), (A
Æ B), (A ´ B), ("xn)(A), ($xn)(B). 6
The usual semantics is in
relational structure is a
is a nonempty set and the
nary relations on D, and
into D. terms of relational structures. A
system M = (D,cn*,Rnm*,Fnm*), where D
cn* are elements of D, the Rnm* are
the Fnm* are nary functions from D To give the usual semantics, we need to use Massignments.
These are functions f from the set of all variables into D =
dom(M).
What we are after is the definition of Sat(M,A,f). I.e., the
formula A holds in the structure M with the Massignment f.
We first have to define Val(M,t,f), where t is a term and f
is an Massignment.
i.
Val(M,xn,f) = f(xn);
ii. Val(M,cn,f) = cn*;
iii. Val(M,Fnm(t1,…,tn)) = Fnm*(Val(M,t1,f),…,Val(M,tn,f)).
We can now define Sat(M,A,f) for atomic formulas A, as
follows.
Sat(M,Rnm(t1,…,tn),f) iff Rnm*(Val(M,t1,f),…,Val(M,tn,f)).
Sat(M,s = t,f) iff Val(M,s,f) = Val(M,t,f).
Finally, we define Sat(M,A,f) for all formulas A as follows.
a.
b.
c.
d.
e. Sat(M,(ÿA),f) iff not Sat(M,A,f);
Sat(M,(A Ÿ B),f) iff Sat(M,A,f) and Sat(M,B,f);
Sat(M,(A ⁄ B),f) iff Sat(M,A,f) or Sat(M,B,f);
Sat(M,(A Æ B),f) iff not Sat(M,A,f) or Sat(M,B,f);
Sat(M,(A ´ B),f) iff (Sat(M,A,f) and Sat(M,B,f)) or (not
Sat(M,A,f) and not Sat(M,B,f));
f. Sat(M,("xn)(A),f) iff for all y Œ A, Sat(M,A,f[n/y]);
g. Sat(M,($xn)(A),f) iff there exists y Œ A, Sat(M,A,f[n/y]).
Here f[n/y] is the same as f except that at xn it is y
(instead of f(xn)).
Let X be a set of formulas and A be a formula. A is valid if
and only if for all structures M and Massignments f,
Sat(M,A,f).
We write Sat(M,X,f) if and only if for all B Œ X, Sat(M,X,f). 7 We say that X logically implies A if and only if for all
structures M and Massignments f, if Sat(M,X,f) then
Sat(M,A,f).
There are nice axioms and rules of inference for predicate
calculus with or without equality. Again we provide a new
twist.
WEAK COMPLETENESS. Every valid formula is provable using
these axioms and rules.
WEAK SOUNDNESS. Every formula provable using these axioms and
rules is valid.
STRONG COMPLETENESS. If a given set of formulas logically
implies a given formula then that formula can be derived form
the set using these axioms and rules.
STRONG SOUNDNESS. If a given formula can be derived from a
given set of formulas using these axioms and rules, then that
set logically implies that formula.
We will give a Hilbert style axiomatization. We use two
rules:
a. Modus ponens. From A and A Æ B derive B.
b. Substitution. From any axiom A derive every substitution
instance of A.
c. Universal generalization. From any axiom A derive ("xn)(A).
Here we have to be much more careful about what we mean by
substitution. I will clarify this carefully later (not
today).
What axioms?
d. All valid formulas with at most ? occurrences of signs
other than parentheses.
I have to experiment to see just what are the most
appropriate forms of d to use.
The point is that very simple formulas for d will do, and one
can then take all formulas of small size. This is now a
recurrent theme in a number of things I work on now. E.g., in 8
Phil 532, I alluded to using this idea to try to uniformly
associate axiomatic theories to concepts or objects.
As opposed to the case with PROP, here we know that there is
no algorithm for determining whether or not a formula is
valid.
The usual move is to look for significant subclasses of
formulas for which there is an algorithm, and then study the
computational complexity of such algorithms. This is normally
done for classes with infinitely many elements. Normally one
runs into recursive unsolvability  no algorithm. A small
dose of flexibility in the formulas will cause this
unsolvability, where nobody knows what to say beyond that.
I have been proposing that one proceed differently as
follows. Look at all kinds of unsolvable classes, including
the entire class of formulas. But don't try to do anything
with the whole class  which you can't. Rather look at
successively larger small initial segments of these classes.
You might say  "but any finite set is recursively solvable,
and so what is there to do?"
But here is the paradigm shift. Deciding membership in a
large finite set is, in actuality, very much like deciding
membership in an infinite set. You want the algorithm to be
of reasonable size. You want to have, as a consequence of
your decision procedure, that every instance of the
membership problem is provably true or provably false in (a
weak fragment of) ZFC (the standard axioms for mathematics).
The fact that the sample set may be finite is no assurance
that this can be done, or that if it can be done, it can be
done easily. In fact, these finite decision problems are
probably much harder than infinite ones that go positively.
Tricky philosophical issues arise regarding just how you want
to say that a large finite set is "intractable". From Gödel,
we know that if you start with any recursively unsolvable set
of finite strings, then for some n, the set of elements of
length at most n is in some sense intractable. For instance,
there is a particular string such that membership in the set
is neither provable nor refutable in ZFC.
2. FORMAL SYSTEMS.
3. CONSISTENCY. 9
4. RELATIVE CONSISTENCY.
5. CONSERVATIVE EXTENSION.
6. INTERPRETATION.
PHILOSOPHY 536
PHILOSOPHY OF MATHEMATICS
LECTURE 2
10/02/02
10/11/02
In this lecture, we discuss formal systems from logic to
Presburger arithmetic.
COMPARISON OF FORMAL SYSTEMS.
In this seminar, we climb up the now standard hierarchy of
formal systems that have emerged in f.o.m. from logic to the
highest of the large cardinal axioms.
By a formal system, we will usually mean a system with the
following ingredients, as discussed in seminar 536, lecture
1.
0.
1.
2.
3.
4.
5. Choice of constants, relations, and functions.
Logical axioms.
Rule of substitution applied to all logical axioms.
Rule of universal generalization, applied to all theorems.
rule of modus ponens, applied to all theorems.
Nonlogical axioms. We consider 1,2,3,4 as fixed, where 1,2 are restricted to the
formulas in the language given by 0. The crucial choice is
that of 0 and 5.
Usually 0 is finite, and often 5 is also finite. When
infinite, it is most often the closure of a finite set under
substitution.
However, some important sets of nonlogical axioms that are
not handled in this way. I am thinking of schemes where
restrictions are placed on the formulas to be substituted
such as on the number of quantifiers, or that all quantifiers
be relativized by some particular formulas, or restrictions
are placed on terms, etc. 10
This flexibility is normally handled by simply requiring that
the set of all nonlogical axioms is "recursively decidable".
However, this is clearly much too general, and one would like
to have a general framework for presenting formal systems
with finitely many nonlogical axiom "schemes", with some
suitably general notion of "scheme".
An attractive approach in many contexts is to bite the bullet
and move over to many sorted predicate calculus, with all of
its complications over single sorted predicate calculus.
Perhaps this is worth doing. Then we can do things like
insist that only quantifiers over certain sorts are present
in a scheme, etc. However, this solves some of the problems
but not all of the problems. That is, the problem of
reflecting that in any reasonable case where infinitely many
nonlogical axioms are used, the nonlogical axioms are
"essentially finite" in number. However, even this move to
many sorted predicate calculus does not handle all useful
cases.
What do we mean by going up or climbing up, as in the first
sentence of this section?
There is a quasi ordering on formal systems, discussed in
536, lecture 1. This is the quasi ordering under
interpretability. (A quasi ordering is a transitive reflexive
relation).
Frequently, but not always, as we go up, we will have that
the weaker system is a subsystem of the stronger system. When
this is not the case, we always have that every sentence of a
certain general kind provable in the weaker system is
provable in the stronger system. The latter will hold in both
directions for systems that are mutually interpretable.
It is nice to know that asserting the interpretability of one
formal system into another is in principle incontrovertible,
in the case of formal systems with finitely many nonlogical
axioms (no nonlogical axiom schemes). This is because such an
interpretation is given by finite data, and asserts the
provability of finitely many statements in the target system.
If we had nonlogical schemes, we are still talking about
finite data, but also we are talking about infinitely many
assertions being provable. 11
The reason it is nice to know that we have
incontrovertibility here is that in general we will be
considering interpretations of one not obviously consistent
system into another not obviously consistent system. The
interpretation of the first into the second establishes that
if the second is consistent then the first is consistent.
Since consistency is the sensitive issue, particularly as we
move up the hierarchy of systems, we would like such a
comparison tool to be as free of controversy as possible.
In the case of where the first formal system is given by
finitely many nonlogical axiom schemes, and not just finitely
many nonlogical axioms, the interpretations, when they exist,
are still satisfactorily incontrovertible in that very little
in the way of epistemic or ontological commitments are needed
in order to establish that one has given an interpretation at least in practice.
Moreover, there appears to be, in practice, quite a robust
associated finitely axiomatized extension of any natural
theory arising with finitely many axiom schemes. This is the
system obtained by adding a new sort for "subsets of the
universe", and adding a comprehension axiom which does not
allow quantification over the new sort. This produces a
formal system extending the original one which, under certain
general conditions, is finitely axiomatizable  and very
demonstrably so. Then one can to some extent have one's cake
and eat it too by always passing to such a finitely
axiomatizable extension when confronted with finitely many
axiom schemes. Use the notation T' for this extension of T.
This immediately leads to the question: what precisely is the
relationship between a given formal system axiomatized by
finitely many axiom schemes, or even an outright finitely
axiomatized system, and T'?
The answer is that we have a conservative extension, in the
sense that every sentence in the language of the original
system is provable in T if and only if it is provable in T'.
However, in general, this requires some epistemic
(ontological?) commitment involving the indefinite iteration
of exponentiation. There is a blowup involved in going from a
proof in T' of a sentence in the original language, to a
proof in T. 12
There appears to be a number of detailed issues concerning
blowups of this kind when passing to extensions of this sort.
One can consider various weaker and stronger such extensions,
with lesser and greater blowups expected. Also, the blowup
one gets in general, or in certain standard cases, may be
greater than the blowups one gets if the original system is
weak; e.g., too weak to interpret a fair amount of formal
arithmetic.
Typically, we will encounter the following situation. We have
two fundamentally important formal systems S,T. We have
i) T proves the consistency of S; or
ii) S proves the consistency of T; or
iii) S,T are "equiconsistent".
As an immediate consequence of i), we have S is interpretable
in T but not vice versa (provided T interprets a reasonable
amount of arithmetic). As an immediate consequence of ii), we
have the other way around: T is interpretable in S but not
vice versa (provided S interprets a reasonable amount of
arithmetic).
The exact meaning of iii) is usually kept fluid. But it
normally means that one can prove with very little epistemic
commitments that
S is consistent if and only if T is consistent.
Often, but not always, we will have that S,T are mutually
interpretable.
We proved a very general theorem to the effect that S,T are
"equiconsistent" if and only if S,T are mutually
interpretable, in the case of finitely axiomatized S,T which
interpret a certain small amount of arithmetic.
The usual statement of this theorem asserts that for finitely
axiomatized systems S,T interpreting a small amount of
arithmetic,
S is inconsistent Æ
T is inconsistent
is provable in a suitably explicit way in a suitably weak
system if and only if 13 S is interpretable in T.
The reverse direction is immediate since interpretations
explicitly map inconsistencies into inconsistencies.
Moreover,
S is inconsistent Æ
T is inconsistent
is provable in a suitably weak system if and only if
S is interpretable in T'.
There are small changes that can be made in the definition of
interpretation  e.g., does equality have to be interpreted
as equality or not, and can objects be interpreted as tuples
of objects, etc. Such theorems as above, relating
interpretability to relative consistency proofs, can be used
to show that under quite general conditions, such small
modifications in the notion of interpretation are
inconsequential.
Recently, we thought about what is the strongest thing that
can be said in the case that S,T are equiconsistent, or
mutually interpretable. We believe that under quite general
conditions, including the situations that we will be
concerned with, S,T are in fact synonymous. Again there are
several notions that differ in detail, but the idea is that
there is a pair of interpretations, one from S into T, one
from T into S, such that the interpretation of an
interpretation of a sentence is equivalent to the sentence
(with the sentence from any of the two sources S,T).
Perhaps this way of talking about synonymy is a bit confusing
and does not directly enough get at the essence of the idea
of synonymy, which is "talking about the same thing in two
different equivalent ways".
It should be possible to make an honest philosophical
analysis of this phrase "talking about the same thing in two
different equivalent ways", and lead to the above formal
notion of synonymy, or at least a closely related one or
ones. 14
The Theorem to look for is that whenever two finitely
axiomatizable theories satisfying quite general conditions
(including interpreting a certain amount of arithmetic) are
mutually interpretable, they are synonymous. The proof would
pass through the relative consistency formulations that we
know are equivalent.
Such results should then be adapted to the situation where we
have finitely many nonlogical axiom schemes.
THE SYSTEM T(=).
Our first system with nonlogical axioms in this Seminar, is
T(=), in the language with just equality. We have the axioms
that assert there are at least n objects, for each n ≥ 2. The
case n = 1 is provable.
Thus the nth axiom is
An = ($x1)...($xn)(B)
where B is the conjunction of all xi ≠ xj, 1 £ i,j £ n.
The most striking thing about the nth axiom is its length.
It has n2 conjunctions of negated equations.
This leads to the following question. Let C be a sentence in
this language which is equivalent to An. What can we say about
the structure of C?
First of all, we can considerably shorten An by writing it in
the logically equivalent form
An = ("x1)...("xn1)($xn)(xn ≠ x1 Ÿ ... Ÿ xn ≠ xn1).
It would appear that An is the simplest way of asserting that
there are at least n distinct objects, in various senses.
Presumably, one can show that one needs at least n
quantifiers. It would also seem that if we use n quantifiers
in prenex form, then the last quantifier must be existential.
In fact, one can study
prenex form, to assert
objects. Note that for
matrix appears to have which quantifier forms can be used in
that there are at least n distinct
some quantifier combinations, the
to be quite long (say with the first 15
An), and with others, the matrix can be comparatively short
(say with the second An).
I am not sure if such a detailed and sharp analysis of
various aspects of this system has been done.
The elimination of quantifiers shows that every formula is
equivalent in T(=) to a formula without quantifiers. The
usual way to prove such things is to show that every formula
starting with a single existential quantifier followed by a
quantifier free part is equivalent in T(=) to a quantifier
free formula.
One then shows that every sentence is equivalent to x = x or
equivalent to x ≠ x. I.e., every sentence is either provable
or refutable in T(=).
In particular, this means that T(=) is a complete theory.
I.e., every sentence in its language can be proved or
refuted. It is provable if and only if it is true in the
infinite structures.
Suppose we are given a sentence with at most n quantifiers.
We should be able to get sharp information on the "size" of a
"smallest" proof of the sentence or its negation. For lower
bounds, we may have to assume the fundamental conjecture on
the lengths of proofs in propositional proof systems (that
they may have exponential size).
It is my impression that there has been definitive work on
the computational complexity of determining whether a
sentence is provable or refutable in T(=), and for several
stronger systems than T(=). But the issue of sizes of proofs
seems to have been neglected.
If a sentence is provable in T(=) then it holds in all
structures of at least some particular size. One should be
able to get sharp information as to the sufficient size in
terms of the number of quantifiers in the given sentence.
More detailed information involving the pattern of
quantifiers in prenex form should also be obtainable.
It would be interesting to present reasonably short provable
sentences whose proofs are all very long. This may or may not
involve solving presently intractable problems related to
computational complexity. 16 THE SYSTEM T(0,S).
Here we will have equality, the constant 0, and the unary
function symbol S. The axioms are:
1. S(x) ≠ 0.
2. S(x) = S(y) Æ x = y.
3. x ≠ 0 Æ ($y)(x = S(y)).
4. S(x) ≠ x.
5. S(S(x)) ≠ x.
...
The "standard" model of T(0,S) is (N,0,S), where S(x) = x+1.
Note that T(0,S) proves T(=). In fact, T(=) is provable with
just 1,2 of T(0,S). To see this, for each n ≥ 0, let n* be
S...S(0), where there are n S's. From n* = m* one gets (n1)*
= (m1)*, etc. If n ≠ m, one eventually gets r* = 0, where r
≠ 0, which is a contradiction.
Again, one has elimination of quantifiers, where every
formula is provably equivalent to a quantifier free formula.
From this, we again get completeness  every sentence is
provable or refutable. We also obtain that a sentence is
provable if and only if it is true in the "standard" model of
T2, which is (N,0,S), where N is the set of all nonnegative
integers and S(x) = x+1.
We can again raise the same issues regarding sizes of proofs.
We can also ask about bounds associated with sentences of the
form
("x1)...("xn)($y)(A(x1,...,xn,y))
that are true in (N,0,S), where A is a formula. The bounds we
are talking about depend on x1,...,xn and the size of A, and
upper bounds the y.
It is clear that every sentence in the language of T(=)
provable in T(0,S) is already provable in T(=). I.e., T(0,S)
is a conservative extension of T(=). This is because if
provable in T(0,S) then it is true in (N,0,S), and hence in
N. Therefore it is provable in T(=). 17
What can we say about the relationship between a proof in
T(=) of a sentence and a proof in T(0,S) of that same
sentence? It might be that the size of some proof in T(0,S)
is significantly shorter than the size of any proof in T(=).
We don't know if this has been thoroughly investigated.
T(0,S) cannot be interpreted in T(=). We can use the
elimination of quantifiers in T0 to show that no relation can
serve as an interpretation of S.
T(0,S) is not finitely axiomatizable. One can show that there
are models of T(0,S) with no loops of size £ n but with a
loop of size n+1, for any given positive integer n.
THE SYSTEM T(0,S,<).
T(0,S,<) has primitives 0,S,<. The axioms of T(0,S,<) are
1.
2.
3.
4. x ≠ 0 Æ ($y)(x = S(y)).
x < S(y) ´ (x < y ⁄ x = y).
ÿx < 0.
< is a linear ordering. The "standard" model of T(0,S,<) is (N,0,S,<).
Note that T(0,S,<) is finitely axiomatizable, unlike T(0,S).
Also note that T(0,S) is a subsystem of T(0,S,<).
Again, we have quantifier elimination, in the usual sense
that every formula is provably equivalent to a quantifier
free formula in T(0,S,<). So we get that every sentence is
provable or refutable in T(0,S,<). This means that a sentence
is provable in T(0,S,<) if and only if it is true in
(N,0,S,<).
Thus a sentence in 0,S, if provable in T(0,S,<), must be true
in (N,0,S), and hence provable in T(0,S). Therefore T(0,S,<)
is a conservative extension of T(0,S).
We can ask our usual questions about lengths of proofs, and
bounds on existential quantifiers.
Is there a significant size reduction for proofs of sentences
in the language of T(0,S) if we allow a proof in T(0,S,<)?
This should be explored. 18
Can T(0,S,<) be interpreted in T(0,S)? The answer is no. Use
the quantifier elimination for T(0,S) to show that no
relation can serve as an interpretation of <.
T(0,S,<) has no finite models. T(0,S,<) has a model whose
complete diagram is recursive  namely (N,0,S,<).
THE SYSTEM T(0,S,+).
T(0,S,+) is Presburger arithmetic. The axioms are rather
long. So we bring in a unifying idea for T(0,S), T(0,S,<),
T(0,S,+).
Let us go back to T(0,S). Consider the following
axiomatization.
1. S(x) ≠ 0.
2. S(x) = S(y) Æ x = y.
3. (A[x/0] Ÿ ("x)(A Æ A[x/S(x)])) Æ A, where A is any
formula in 0,S.
Thus we are using the induction scheme for all formulas in
the language of T(0,S).
It can be easily verified that T(0,S) is a subsystem of 13.
However, every sentence provable in 13 is true in (N,0,S),
and therefore provable in T(0,S). Hence 13 is an
axiomatization of T(0,S).
Now go back to T(0,S,<). Consider the following
axiomatization.
1. S(x) ≠ 0.
2. S(x) = S(y) Æ x = y.
3. x < S(y) ´ (x < y ⁄ x = y).
4. ÿx < 0.
5. (A[x/0] Ÿ ("x)(A Æ A[x/S(x)])) Æ A, where A is any
formula in 0,S,<.
It can be easily verified that T(0,S,<) is a subsystem of 15.
Of course, since T(0,S,<) is finitely axiomatized, something
is lost in using the axiomatization 15. 19
An interesting question is to what extend are proofs
shortened by using the induction scheme both in the case of
T(0,S) and in the case of T(0,S,<).
We now use this unifying induction idea to give an
axiomatization of what we call Presburger arithmetic, or in
our notation, T(0,S,+). The axioms of T(0,S,+) are
1. S(x) ≠ 0.
2. S(x) = S(y) Æ x = y.
3. x + 0 = x.
4. x + S(y) = S(x + y).
5. (A[x/0] Ÿ ("x)(A Æ A[x/S(x)])) Æ A, where A is any
formula in 0,S,+.
In the version with 0,S,<,+, we simply add the definition of
< as follows:
6. x < y ´ ($z)(z ≠ 0 Ÿ x + z = y).
We write this as T(0,S,+,<).
The elimination of quantifiers for Presburger arithmetic is
more difficult than for T(=), T(0,S), T(0,S,<). First of all,
it is false if taken literally  i.e., that every formula in
0,S,+ is equivalent to a quantifier free formula in 0,S,+, or
even that every formula in 0,S,+,< is equivalent to a
quantifier free formula in 0,S,+,<.
The correct elimination of quantifiers must be done for an
expanded language. The expanded language consists of 0,S,<,+,
and for each n ≥ 2, a binary relation ≡n, whose intended
meaning is "congruent modulo the standard integer n". Here we
view ≡n as defined by
x ≡n y ´ ((x £ y Ÿ ($z)(y = x + nz)) ⁄ (y < x Ÿ ($z)
(x = y + nz)))
where nz abbreviates z + ... + z, where there are n z's.
The background fact is that T(0,S,+) proves the crucial
quotient remainder theorem, which asserts that for all
standard integers n ≥ 2,
($y)($z)(x = ny + z Ÿ z < n) 20
where again ny is y + ... + y, where there are n y's.
This expansion of the language is by definable relations. So
it is clear what we mean by quantifier elimination for
Presburger arithmetic for this language.
As a Corollary to the quantifier elimination, we get that
every sentence of T(0,S,+) is provable or refutable (for any
of the languages under consideration).
We ask our usual questions about bounds and sizes of proofs.
Also to what extent proofs are shortened by using T(0,S,+,<)
over the different axiomatizations of T(0,S,<).
It is possible to give a much more explicit set of
mathematically clear axioms for T(0,S,+) and T(0,S,+,<), and
even T(+), although this is at the cost of having quite a
large number. Presumably axioms with at most one existential
quantifier will suffice (with free variables allowed). The
axioms can be given in the following form: the quotient
remainder theorem as a scheme for each divisor ≥ 2, together
with finitely many axioms.
Presburger arithmetic represents a very significant portion
of important mathematics with a complete axiomatization,
where the axiomatization is by finitely many axiom schemes in
the usual sense. "Integral semilinear geometry".
PHILOSOPHY 536
PHILOSOPHY OF MATHEMATICS
LECTURE 3
10/16/02
1. Structure of definable structures.
We first discuss a somewhat new kind of problem for
structures. We have considered a number of structures where
the definable sets are well behaved; e.g.,
(N,=), (N,0,S), (N,0,S,<), (N,<), (N,+)
Last lecture, we gave complete axiomatizations for all of
them except (N,<). This can be given in terms of the complete
axiomatization given for (N,0,S), since 0,S are definable in
(N,<). We have quantifier elimination for the first three
structures without expanding the language. We have quantifier 21
elimination for the last two structures in a suitably
expanded language.
We can ask for rather strong information about the definable
sets. In one dimension, the definable sets are particularly
simple, and the kind of information we will talk about
doesn't surface. The family of definable subsets of N in the
five structures above are, respectively,
i) (N,=). The finite and cofinite sets;
ii) (N,0,S). The finite and cofinite sets;
iii) (N,0,S,<). The finite and cofinite sets;
iv) (N,<). The finite and cofinite sets.
v) (N,+). The ultimately periodic sets.
Let M be a structure. By an Mdefinable set we will always
mean a multidimensional Mdefinable set.
We can consider all structures that are Mdefinable. This
means a structure in a finite relational type whose domain is
an Mdefinable set, and whose relations and functions are Mdefinable. In this context, the relations and functions are
treated as multidimensional sets in the relevant dimension.
The strong info that we are talking about consists of getting
detailed information about the Mdefinable structures up to
isomorphism.
To illustrate the depth of this problem even for (N,<), let
alone (N,+), consider just linear orderings. Some of them are
well orderings. E.g., we have the lexicographic orderings x <k
y ´ (x,y Œ Nk Ÿ (xi < yi, where i is least such that xi ≠
yi)).
The decidability of whether
ordering definable in (N,<)
decidability of the monadic
monadic second order logic,
predicate calculus, but one
the domain. or not one has presented a well
follows from the known
second order theory of (N,<). In
one has the usual first order
can also quantify over subsets of However, the monadic second theory of (N,+) is undecidable,
since multiplication can be monadic second order defined. So
you have to go back to the salt mines to prove the
decidability of whether or not one has presented a well
ordering definable in (N,+). 22 In any case, one wants to prove that the sup of the well
orderings definable in (N,<), or even (N,+), is ww. This
should be true.
PROPOSITION 1.1. The monadic second order theory of (N,+) is
undecidable.
Proof: To see this, first recall that < is definable in
(N,+). Let E be the set of all squares. We claim that the set
of first order sentences true in (N,+,E) is undecidable. This
is because squaring can be defined in (N,+,E) by
r = n2 ´ r Œ E and the next element of E is r+2n+1.
Also multiplication can be defined from squaring by
t = nm ´ 2t = (n+m)2  n2  m2.
We claim that E is the unique solution to a first order
predicate y(E) in (N,+). We just say that 0,1 Œ E, and for
all successive n < m < r from E, mr = mn+2.
We can convert any first order sentence j in (N,+,E) to a
monadic second order sentence j* in (N,+), such that j is
true in (N,+,E) if and only if j* is true in (N,+), by taking
j* = ("E)(y(E) Æ (N,+,E) satisfies j). But we can convert
any first order sentence in (N,+,•) to a first order sentence
in (N,+,E). QED
I can show that the problem of whether or not one definable
structure over (N,<) is embeddable into another is
undecidable. In fact, I am working on a proof that this
problem is complete S11.
I lean towards the opinion that the problem of whether or not
two definable structures over even (N,<) are isomorphic is
undecidable. I'm working on a proof of this, using the
undecidability of Hilbert's 10th problem. This should be
easier to pull off for (N,+).
2. Complexity of axiomatizations.
A second somewhat new kind of problem relates to the
complexity of axiomatizations. The aim is to find the
"simplest" axiomatization, or "a simplest" axiomatization of 23
familiar theories. We have already touched on this earlier,
in connection with the complexity of sentences equivalent to
"there are at least k objects".
For instance, one can look at the total number of quantifiers
used for an axiomatization with sentences, and attempt to
minimize it. Of course, even more delicate is issues
surrounding the pattern of universal and existential
quantifiers in prenex form.
As an example, let us consider the axioms for strict linear
ordering, as sentences. In the usual axiomatization, we can
combine universal quantifiers to get an axiomatization using
3 universal quantifiers. Presumably it is a theorem that one
needs 3 quantifiers. In fact, that one needs 3 universal
quantifiers, when restricting attention to prenex sentences.
Recall the finite complete axiomatization we discussed of
(N,0,S,<). Even more elemental is the complete axiomatization
of (N,<) that can be derived from it. It would appear that 4
quantifiers are necessary for this, including, in the prenex
case, three universal and 1 existential quantifier.
One can continue this study completely systematically.
3. Axiomatizations for the rationals.
(Q,<) is completely axiomatized by the axioms of dense linear
order without endpoints. This has quantifier elimination.
Presumably, (N,<) and (N,0,S) are not isomorphic to definable
structures over (Q,<), and (Q,<) is not isomorphic to a
definable structure over (N,0,S,<), or even over (N,+). I am
a little bit queasy about this second statement, though.
In any case, it seems almost clear that formal system T(Q,<)
is not interpretable in T(N,+), and T(N,0,S) is not
interpretable in T(Q,<).
We now consider (Q,+). This also has quantifier elimination,
without going to an expanded language. This is different than
the case of (N,+), where in order to have quantifier
elimination, we must introduce infinitely many new unary
relations. It seems likely that (Q,+) is not isomorphic to
any definable structure over (N,+), and vice versa. Also, it 24
seems almost clear that T(Q,+) and T(N,+) are not
interpretable in each other.
Now consider (Q,+,<). Obviously, < is definable in (N,+).
However, < is not definable in (Q,+). We have quantifier
elimination for (Q,+,<) without expanding the language.
Presumably, (Q,+,<) is not isomorphic to any structure
definable in (Q,+), and T(Q,+,<) is not interpretable in
T(Q,+).
The one dimensional definable sets in (Q,<) are exactly the
finite unions of intervals, where the intervals can have
endpoints on one side or the other, or be infinite on one
side or the other. This is also true of (Q,+,<).
In (Q,+), the one dimensional definable sets are finite or
cofinite.
5. Axiomatizations for the integers.
Just as we have considered, (N,=), (N,0,S), (N,0,S,<), (N,<),
(N,+), we can also consider (Z,=), (Z,0,S), (Z,0,S,<),
(Z,<),(Z,+),(Z,+,<).
All of these have simple complete axiomatizations. The one
for (Z,=) is the same as the one for (N,=) since they are
isomorphic.
For (Z,0,S), we can take two numbers have the same successors
iff they are equal, no loops, and every number has a
predecessor. The constant 0 serves no purpose, and so we can
use (Z,S) instead, and have quantifier elimination. Note that
(N,S) does not have quantifier elimination without adding,
say, 0.
For (Z,0,S,<), we have a simple finite axiomatization, and
also quantifier elimination. Unlike the case for N, we don't
need 0 for the quantifier elimination.
For (Z,<), we can get an axiomatization via (Z,S,<).
(Z,+) does not admit quantifier elimination unless we add
infinitely many new unary predicates for divisibility, as in
the case of (N,+). Only in the case of (N,+), we also had to 25
add <. Here we do not need <, and in fact < is not definable
in (Z,+). Moreover, N is not definable in (Z,+).
(Z,+,<) admits quantifier elimination if we add infinitely
many new unary predicates for divisibility, and the complete
axiomatization can be obtained from that for (N,+,<).
6. Axiomatizations for the reals.
Here we can go much further, and bring in multiplication.
Let's proceed in steps.
(R,<) has quantifier elimination, and (R,<),(Q,<) are
elementarily equivalent; i.e., satisfy the same sentences. In
fact, (Q,<) is an elementary substructure of (R,<). This
means that any formula with parameters from Q holds in (Q,<)
if and only if it holds in (R,<).
(R,+) has quantifier elimination, and (Q,+) is an elementary
substructure of (R,+).
(R,+,<) has quantifier elimination, and (Q,+,<) is an
elementary substructure of (R,+,<). The one dimensional
definable sets are exactly the finite unions of intervals.
We now come to a main event. (R,+,•) has quantifier
elimination. We have to use the expanded language
(R,<,+,•,0,1). Of course, < is definable in (R,<,+,•,0,1),
but not in (R,+).
I have my own pet treatment of this quantifier elimination,
but I asked Dave Marker at University of Illinois at Chicago
for his favorite references. He responded:
The book "Real Algebraic Geometry" by Bochnak, Coste and Roy
presents a geometric proof of quantifier elimination for real
closed fields.
The book "Quantifier Eliination and Cylindric Algebraic
Decomposition", SpringerVerlag, 1998. B. F. Caviness and J.
R. Johnson (eds.), is a "reader" on qe. It contains original
papers by Tarski, Collins paper describing cylindric
decomposition and some of the more modern papers in the
subject. 26
In my Model Theory of Fields book I prove quantifier
elimination using model theory and the Robinson/Blum style
methods for proving qe from embedding results. I am in the
process of writing a introductory model theory text book.
Chapter 3 discusses the model theoretic approach to qe and
studies algebraically closed and real closed fields.
A first draft of the book is available on [Marker's] web
page:
http://www.math.uic.edu/~marker/CIMT901.ps
I think Marker's book is now out, published by Springer.
As a consequence of the quantifier elimination for
(R,<,+,•,0,1), we get that the one dimensional definable sets
are again just the finite unions of intervals.
There are a number of important complete axiomatizations for
(R,<,+,•,0,1). The first one is given by the so called "real
closed ordered field" axioms, written RCOF:
A.
1.
2.
3.
4.
5.
6.
7.
8.
9. Ordered field axioms.
+ is commutative and associative.
• is commutative and associate.
x(y+z) = xy + xz.
x+0 = x, x•1 = x.
< is a strict linear ordering, 0 < 1.
x < y Æ x+z < y+z.
(x < y Ÿ 0 < z) Æ xz < yz.
($y)(x + y = 0).
x ≠ 0 Æ ($y)(x•y = 1). B. Completeness axioms.
10. Every nonnegative element has a square root.
11. Every polynomial of odd degree has a root.
Note that 11 is an axiom scheme, which can be written as
($x)(xn + an1xn1 + ... + a1x + a0 = 0)
where n is an odd positive integer.
Obviously RCOF holds in (R,<,+,•,0,1). The only issue is 11,
but since n is odd, we see that the polynomial must take on a
negative value for negative enough x, and a positive value
for positive enough x. So by the intermediate value theorem,
it has a root. 27 That, however, is a consistency proof using ideas embedded in
real analysis. We address the issue of consistency proofs
later.
Another axiomatization of RCOF is as follows.
A. Ordered field axioms.
B. Intermediate value. Every polynomial that takes on a
negative value at the left endpoint of a closed interval, and
a positive value at the right endpoint of that closed
interval, has a zero in the interior of that closed interval.
It is obvious that the first axiomatization is a subsystem of
this second axiomatization. The reverse follows the
completeness (discussed below), but one would like a more
elementary proof of the logical equivalence of these two
systems. This can be done.
We now give another kind of axiomatization.
A. Ordered field axioms.
B. Least upper bound scheme. For any formula in the language,
with one free variable and parameters allowed, there is a
least nonnegative element which is at least as large as any
solution to the formula.
It is clear how the earlier axiomatizations of RCOF are a
subsystem of this axiomatization, but again the obvious proof
of the reverse goes through completeness. One can get an
elementary proof of the logical equivalence of all three
axiomatizations.
We now take up the matter of axiomatizations that do not use
<. I.e., just for (R,+,•,0,1).
The most well known one is the so called axioms for real
closed fields (RCF).
A. Field axioms.
B. Realness. 1 is not the sum of finitely many squares.
C. Square roots. For all x, x has a square root or x has a
square root.
D. Zeros. Every polynomial of odd degree has a root. 28
The above is the most "bare bones" axiomatization known of
(R,+,•,0,1). But what does that mean?
It is easy to see that this latter axiomatization is complete
for (R,+,•,0,1) if and only if the first axiomatization is
complete for (R,<,+,•,0,1).
It is easy to show that realness is a consequence of the
first axiomatization, since squares can be proved to be
nonnegative. So the first axiomatization is, from the
algebraic point of view, simply the real closed field axioms.
The normal way of doing the quantifier elimination for the
first axiomatization, and thereby deriving completeness as a
consequence, is to first derive the second axiomatization.
However, this is normally done purely algebraically, thereby
passing through the completeness theorem for predicate
calculus. One goes through the Artin theory of real closed
fields. One first proves that if you adjoin sqrt(1) to a
real closed field, then the field becomes algebraically
closed. Then one has the theory of algebraically closed
fields available, including factoring of polynomials. One
then shows that every polynomial over the real closed field
can be factored into linear and quadratic factors. From
there, one derives the intermediate value property for
polynomials.
After this step, the quantifier elimination is fairly
straightforward, but still has to be done carefully.
7. Conservative extension.
There is a very important conservative extension result.
THEOREM 5.1. Tarski?. Every purely universal sentence
provable in RC0F is provable in OF (the ordered field
axioms).
I have not seen a careful study of the blowups involved in
Theorem 5.1. However, it appears that they are roughly on the
order of a double exponential.
Normally this is proved algebraically via the classical
result that every real field can be embedded in a real closed
field. This proof does not readily give any bounds. 29
It would be very interesting to give a set of striking simple
examples of sentences in RC0F whose proof in RC0F is far
shorter than its proof in OF.
8. Consistency proof.
We have given a detailed sketch of a consistency proof of all
of the real closed (ordered) field axioms within a weak
fragment of arithmetic. The most natural fragment to do such
a consistency proof is the system EFA of exponential function
arithmetic. The proof is a somewhat big deal, requiring novel
formalizations of classical algebraic arguments.
9. A combined system.
Consider (R,Q,Z,<,+). This is well behaved, with a complete
axiomatization, and quantifier elimination, provided one adds
divisibility predicates.
10. An axiomatization of euclidean plane geometry based on
distance comparison.
In the algebraic approach to geometry, one defines the plane
and its Euclidean metric in the usual first order way in the
field of real numbers. In this way, the first order Euclidean
plane geometry reduces to the first order algebra of real
numbers.
Under the geometric approach, we consider structures such as
(R2,E), where E(x,y,z,w) holds if and only if d(x,y) = d(z,w);
i.e., the Euclidean distance between x and y is the same as
the Euclidean distance between z and w.
We can define the field of real numbers in (R2,E). This is
done by equivalence classes of pairs of elements of R2.
In fact, much more is true. This is almost surely known  but
I don't know a reference.
THEOREM 10.1. The sets definable in (R2,E) are exactly the
semialgebraic subsets of R2. By Tarski, these are also the
subsets of R2 definable in (R,0,1,+,•). The sets definable in
(R2,E) with only the three parameters (0,0),(0,1),(1,0) are
exactly the semialgebraic subsets of R2 presented with
rational coefficients, or the subsets of R2 that are
0definable in (R,0,1,+,•). 30
(Here definable means definable with any number of
parameters. And 0definable means definable with no
parameters).
We also consider interpretability.
THEOREM 10.2. (R,0,1,+,•) is interpretable in (R2,E) and vice
versa.
Here, interpretations are allowed to be via equivalence
relations of tuples, which is allowed in a well known general
form of interpretability.
According to Tarski, the first order theory of (R,0,1,+,•)
has a beautiful axiomatization via the real closed field
axioms:
1)
2)
3)
4) the usual field axioms;
1 is not the sum of squares;
for all x, x or x is a square;
every polynomial of odd degree has a root. Tarski showed that these axioms are complete. Thus a sentence
is true in (R,0,1,+,•) if and only if it is derivable from
these axioms.
Think of (R2,E) as corresponding to the geometric approach to
Euclidean plane geometry, and (R,0,1,+,•) as corresponding to
the algebraic approach to Euclidean plane geometry.
The following question arises. Can we give a similarly
elegant and basic axiomatization of the first order theory of
(R2,E) involving only R2,E?
We also consider a related matter. According to Tarski, the
real algebraic numbers are the 0definable elements of
(R,0,1,+,•). But they have a very algebraic definition (hence
the name "real algebraic"): the solutions to nontrivial
polynomials in one variable with integer coefficients. In
fact, this is the usual definition of real algebraic numbers.
Fundamental to all aspects of this theory is the concept of
**equality condition**. An equality condition is a formula in
the language of (R2,E) of the following special form: a
conjunction of one or more atomic formulas of the form
E(x,y,z,w), where x,y,z,w are variables. 31 We can think of an equality condition in variables x1,...,xk
as a Euclidean plane geometric diagram. There are k labeled
points x1,...,xk in the diagram. There is an indication that
for various pairs of points x1,...,xk, we have equality of
distance. Thus we could mark the line segments joining
various pairs in such a way that pairs that are to have
the same distance have their line segments marked with the
same marking.
Of course, here one must be entirely noncommittal about
degeneracies; e.g., about which of the x's are equal or
unequal, which line segments cross or don't cross, which
triples of points are or are not colinear, etc.
THEOREM 3. Let x Œ R2. The following are equivalent:
a) x is definable in (R2,E) with parameters (0,0),(1,0),(0,1);
b) x is 0definable in (R,0,1,+,•);
c) x is real algebraic (i.e., its components are real
algebraic numbers);
d) x is a coordinate in some solution of some equality
condition in
(R2,E) with parameters (0,0),(1,0),(0,1) that has at most
finitely many solutions;
e) x is a coordinate in some solution of some equality
condition in (R2,E) with parameters (0,0),(1,0),(0,1) all of
whose solutions are permutations of each other.
We can alter the notion of equality condition to incorporate
various commitments. E.g., we can consider modified equality
conditions which consist of a conjunction of one or more
atomic formulas without equality and the conjunction
asserting that all points used are distinct. This eliminates
the most basic of degeneracies, although there are other
important degeneracies still allowed. Then Theorem 3 still
holds.
Furthermore, various other modifications with regard to the
elimination of degeneracies can be made, with the same
result.
We now come to the more delicate matter of the complete
axiomatization of geometry in terms of diagrammatic axioms.
For this purpose it is very convenient to use the 4ary
relation LE(x,y,z,w) on R2, meaning that the distance between 32
x,y is less than or equaled to the distance between z,w. This
relation can be defined nicely from E(x,y,z,w). We give such
a definition below.
First of all, we want to nicely define the midpoint between
two points x,y. Note that there are points z,w (or w,z), such
that x,y,z,w forms a square with diagonal x,y (degenerate if
and only if x = y). This is defined using equidistance  the
sides are all equal and the two diagonals are equal. The
unique point equidistant to the four corners is the desired
midpoint.
Now we are in a position to define LE(x,y,z,w) as follows.
Let p be the midpoint between z,w. Then LE(x,y,z,w) if and
only if there exists u such that d(x,u) = d(u,y) = d(z,p).
Since we are now admitting the ordering directly, it is
appropriate to consider ordered fields and real closed fields
using <. We assume familiarity with the usual ordered field
axioms. In this context, the real field axiom 2) is
superfluous. Thus in this context, real closed fields are
given by
1) the usual ordered field axioms;
2) every all x >= 0 has a square root;
3) every polynomial of odd degree has a root.
It is very useful to separate out what we call basic
Euclidean plane geometry and quadratic Euclidean plane
geometry.
Basic Euclidean plane geometry consists of the set of all
sentences of (R2,LE) that become provable from the axioms of
ordered fields under the obvious translation of (R2,LE) into
(R,<,0,1,+,•). The quadratic ordered field axioms consist of
just axioms 1)  2) above. Quadratic Euclidean plane geometry
consists of the set of all sentences of (R2,LE) that become
provable from the quadratic real ordered axioms under the
obvious translation of (R2,LE) into (R,0,1,+,•).
One can give reasonably elegant axiomatizations of basic
Euclidean plane geometry and quadratic Euclidean plane
geometry staying within the language of (R2,LE). The latter
corresponds closely to ruler and compass constructions. This
much is well known. 33
We now come to the main issue of giving a geometric form of
axiom scheme 3)  that every polynomial of odd degree has a
root  within the language of (R2,LE).
We certainly don't want to simulate this axiom scheme 3)
directly. E.g., odd degree appears to be geometrically
meaningless. But we are looking for a kind of geometric
construction principle.
For this purpose, we define a comparison condition as the
conjunction of finitely many atomic formulas LE(x,y,z,w),
where x,y,z,w are variables.
Let j(x1,...,xn,y1,...,ym) be a comparison condition, where
n,m ≥ 0.
I. Suppose j(x1,...,xn,y1,...,ym). Then we can adjust y1,...,ym
so that j(x1,...,xn,y1,...,ym) and the maximum distance from x1
to the points y1,...,ym is minimized.
Think of x1 as the center of a closed disk in which the new
points y1,...,ym to be constructed are to lie. We want to
minimize the radius of that closed disk.
THEOREM 4. A sentence is true in (R2,LE) if and only if it is
provable from the axioms of basic Euclidean geometry plus the
axiom scheme I.
There are a number of variants of * where related quantities
are minimized. Here are some examples.
II. Suppose j(x1,...,xn,y1,...,ym). Then we can adjust
y1,...,ym so that j(x1,...,xn,y1,...,ym) and the maximum
distance from x1 to the points x1,...,xn,y1,...,ym is
minimized.
III. Suppose j(x1,...,xn,y1,...,ym). Then we can adjust
y1,...,ym so that j(x1,...,xn,y1,...,ym) and the diameter of
y1,...,ym is minimized.
IV. Suppose j(x1,...,xn,y1,...,ym). Then we can adjust
y1,...,ym so that j(x1,...,xn,y1,...,ym) and the diameter of
x1,...,xn,y1,...,ym is minimized.
Theorem 4 holds for any of I  IV. 34
PHILOSOPHY 536
PHILOSOPHY OF MATHEMATICS
LECTURE 4
10/23/02
We have finished discussion of the tame systems, and we now
take up the interval from Presburger arithmetic to Peano
arithmetic.
The best source for this material is the book
Peter Hajek, Pavel Pudlak, Metamathematics of FirstOrder
Arithmetic, Springer, 1998.
1. The main language, and formulas.
The main language that we use is L0 = 0,S,+,•,£,=. We use the
abbreviations
x
x
x
x <
≥
>
≠ y
y
y
y ´
´
´
´ x £ y Ÿ ÿx = y.
x £ y.
y > x.
ÿx = y. Let S0 = P0 the class of bounded formulas, which are the
formulas in L whose quantifiers are bounded; i.e., whose
quantifiers are of the form
("x £ y), ($x £ y)
where x,y are distinct variables, and this is expanded out
using the definition of £.
Sn+1 is the class of formulas ($x)(j), where j is Pn. Pn+1 is
the class of formulas ("x)(j), where j is Sn.
We always let n* = SS...S(0), where there are n S's.
2. R.M. Robinson's system Q.
The language is 0,S,+,•,=. The axioms are
1.
2.
3.
4. S(x) ≠ 0.
S(x) = S(y) Æ x = y.
x ≠ 0 Æ ($y)(x = S(y)).
x+0 = x. 35
5.
6.
7.
8. x+S(y) = S(x+y).
x•0 = 0.
x•S(y) = (x•y)+x.
x £ y ´ ($z)(x+z = y). THEOREM 1.1. A S0 sentence is true iff it is provable in Q.
Proof: By induction on the S0 sentence. This needs some basic
facts about £ provable in Q. E.g., x £ n iff x = 0 ⁄ ... ⁄ x
= n*. QED
THEOREM 1.2. Q is essentially undecidable. I.e., no
consistent extension of Q is decidable.
Proof: Let A,B be two r.e. recursively inseparable sets. Let
($x)(j(x,y)) define A and ($x)(y(x,y)) define B, where j,y
are bounded. Consider a(y) = ($x)(j(x,y) Ÿ ("z £ x)
(ÿy(x,z))), b(y) = ($x)(y(x,y) Ÿ ("z £ x)(ÿj(x,z)). These
still define A,B, and also for every n ≥ 0, Q proves ÿ(a(n*)
Ÿ b(n*)). Let T be a consistent extension of Q. Then {n: T
proves a(n*)} contains A and is disjoint from B. Therefore it
is not recursive. QED
Q was set up to be a kind of minimal natural finitely
axiomatized system in L0 which is essentially undecidable. Can
we make sense of "minimal natural" here?
Q is known not to prove the commutativity of +. Presumably it
cannot prove the associativity of +, the commutativity of •,
the associativity of •, the transitivity of £, the
connectedness of £.
It is perhaps somewhat surprising that stronger than expected
extensions of Q are interpretable in Q.
An appropriate form of Gödel's second incompleteness theorem
can be given for consistent extensions of Q. One needs only a
S1 definition of the set of all formulas provable in T
satisfying the provability conditions.
3. Iopen.
Iopen is Q together with the induction scheme
(j[x/0] Ÿ ("x)(j Æ j[x/S(x)])) Æ j 36
where j is a formula in L0 without quantifiers.
This definitely allows us to go much farther in terms of
elementary mathematical theorems. E.g., the axioms of
discretely ordered commutative semiring with unit. Also, the
appropriate quotient remainder theorem. So we can formally
construct the discretely ordered commutative semiring with
unit of integers, with its quotient remainder theorem. Iopen
proves the existence of a very specific oneone onto
quadratic pairing function (using multiplication by 1/2).
I don't know of any interesting theorem that characterizes
what is provable in Iopen in an interesting class of
sentences. For example, what can we say about the class of
Diophantine equations that can be proved to have no solutions
in Iopen? Is it even decidable? This might be known.
It is known that Iopen does not prove ("x)(x•x ≠ 2), and does
not prove Fermat's last theorem for exponent 3. Such
independence results are proved via the proof of the
following.
THEOREM 3.1. Iopen has a recursive nonstandard model.
I.e., one whose domain is w, and where S,+,•,£ are recursive.
Presumably, Iopen does not suffice to prove FLT for any given
exponent. Also presumably not the existence of a gcd, or that
if x,y ≥ 1 are relatively prime, then we can write nx + my =
1, where n,m are integers.
Because of the appropriately provable pairing function, we
see that for any extension T of Iopen (in L0), the formulas
that are provably equivalent in T to Sn (Pn) formulas are
closed under disjunction, conjunction, and existential
(universal) quantification.
Iopen is interpretable in Q.
4. IS0.
IS0 is Q together with
(j[x/0] Ÿ ("x)(j Æ j[x/S(x)])) Æ j 37
where j is a S0 formula. This is also called polynomially
bounded arithmetic.
IS0 is sufficient to prove that there is no square root of 2,
the existence of gcd's, and the basic theory of gcd's and
lcm's.
Also, IS0 is sufficient to prove that a specific S0 formula of
three variables acts as the graph of the binary exponential
function, which the usual elementary properties and
inequalities, but without proving that exponentiation defined
this way is total. That is known to be an impossible
additional requirement. IS0 can prove that any two such S0
formulas must be provably equivalent.
In the same vein, assuming that nn exists (which can be
formalized using the previous paragraph), we have a coding
system for length £ n sequences from [0,n] with all of the
usual properties. Also assuming that nn exists, IS0 is
sufficient to state and prove the fundamental theorem of
arithmetic on [0,n].
Presumably, IS0 is sufficient to prove a lot of number theory
on [0,n], assuming nn exists. E.g., between any positive
integer and its double (closed interval) there is a prime.
Also, presumably, FLT with exponent n on [0,n], assuming nn
exists. In fact, these are strong forms of a conjecture that
we later make about IS0 + exp and IS0(exp).
THEOREM 4.1. There is no recursive nonstandard model of IS0.
In fact, in any nonstandard model M of IS0 whose domain is a
subset of w, whose addition and multiplication are not
recursive.
Somewhat surprisingly, IS0 is interpretable in Q.
THEOREM 4.2. Every P2 sentence provable in IS0 has a Skolem
function that is everywhere bounded by a polynomial.
We can formulate IS0 using the least number principle instead
of induction. We get a logically equivalent system. The same
is true of order induction (course of values induction).
5. IS0(exp) and IS0 + exp. 38
The system IS0(exp) is defined by first expanding the language
L0 with S,+,•,£,= to L0(exp) with S,+,•,exp,£,=. The S0(exp)
formulas are defined the same way as the S0 formulas; the
formulas of L0(exp) with all quantifiers bounded (to
variables).
The axioms of IS0(exp) consist of Q, induction with respect to
all formulas in L0(exp), and the axioms for exponentiation,
20 = S(0), 2S(x) = 2x + 2x.
The system
defined in
above that
Recall its IS0 + exp of course is in the language L0. Exp is
terms of the formula of three variables discussed
codes the exponential function, provably in IS0.
uniqueness property provably in IS0. THEOREM 5.1. IS0(exp) and IS0 + exp prove the same sentences
in L0.
Here is a nonobvious fact about this system.
THEOREM 5.2. IS0(exp) and IS0 + exp are finitely
axiomatizable.
In IS0(exp), we can freely do finite sequence coding without
worrying about the existence of exponentials. In a sense,
IS0(exp) appears to be the weakest system with genuine
metamathematical freedom.
IS0(exp) is an extremely powerful theorem from the point of
view of finite mathematics. I have conjectured that every
published finite theorem highly valued within the current
core mathematical culture, with a robust formulation in
L0(exp), is provable in IS0(exp).
This conjecture would encompass things like Wiles' FLT,
Falting's Mordell conjecture, and many other landmarks.
However, one theorem that is definitely not provable there is
the following.
THEOREM 5.2. The finite Ramsey theorem is unprovable in
IS0(exp). 39
By this I mean that every sufficiently large coloring of the
unordered n tuples by r colors has a monochromatic set with p
elements. I.e., sufficiently large relative to p,r.
The reason for this is the following.
THEOREM 5.3. Every P2 sentence provable in IS0(exp) has a
Skolem function that is everywhere bounded by an iterated
exponential.
THEOREM 5.4. The finite Ramsey theorem is a P2 sentence that
does not have a Skolem function that is everywhere bounded by
an iterated exponential.
THEOREM. 5.5. The finite Ramsey theorem is a P2 sentence that
is not provable in IS0(exp).
Another important theorem of a metamathematical nature that
is not provable in IS0(exp) is cut elimination.
THEOREM 5.6. The cut elimination theorem for predicate
calculus is a P2 sentence that does not have a Skolem
function that is everywhere bounded by an iterated
exponential.
THEOREM 5.7. The cut elimination theorem for predicate
calculus is a P2 sentence that is not provable in IS0(exp).
In fact, we can be more specific in the form of a reversal.
Let superexp be the sentence in L0(exp) that asserts that for
all n,k, the kfold exponential to base 2 of n exists. This
is formulated in terms of finite sequences, which we are now
comfortable with in IS0(exp).
Alternatively, we could formulate superexp as a sentence in
L0. In fact, we can consider the obvious systems IS0 +
superexp, IS0(exp) + superexp, and IS0(superexp). Then all
three systems prove the same sentences in L0.
THEOREM 5.8. The following are provably equivalent in
IS0(exp).
i) finite Ramsey theorem;
ii) cut elimination for predicate calculus;
iii) superexp. 40
THEOREM 5.9. IS0(exp) is not interpretable in Q.
Here is an important theorem of Wilkie.
THEOREM 5.10. Let A be a P1 sentence in L(exp). Then IS0(exp)
proves A iff Q + A is interpretable in Q.
THEOREM 5.11. IS0(exp) does not prove Con(Q). IS0(exp) proves
cut free Con(Q).
THEOREM 5.12. IS0(exp) + superexp proves Con(IS0(exp)).
The second incompleteness theorem for consistent extensions
of Q works for cut free consistency.
Under very general conditions, if one system proves the
consistency of another, then the former cannot be interpreted
in the other. Therefore IS0(exp) + superexp is not
interpretable in IS0(exp).
IS0(exp) is exactly the right place in arithmetic to be
developing the theory of finite sets of natural numbers. One
can easily prove all of the elementary facts about them,
based on any reasonable coding of them as natural numbers. In
particular, one can handle union, intersection, set
difference, Cartesian product, sum set, and power set, and
appropriately formalize and prove that the set of all (codes
for) subsets of [1,n] has cardinality 2n. The same remarks
apply to the elementary theory of relations, finite
sequences, finite functions, domains, ranges, etc.
If we formulate IS0(exp) with the least number principle
instead of the axiom of induction, then we get a logically
equivalent system. The same is true of order induction
(course of values induction).
In fact, we have worked on set theories that correspond to
IS0(exp) and some weaker fragments as well. These set theories
all have the first w stages of the cumulative hierarchy as
their intended interpretations; i.e., V(w).
6. PRA.
PRA is primitive recursive arithmetic. There are a few
formulations. All of the ones that we consider have the
axioms for successor, and the defining equations. 41 The language consists of =,0,S, and symbols for each
primitive recursive function as they are introduced by
defining equations.
1. S(x) ≠ 0, S(x) = S(y) Æ x = y.
2. Z(x) = 0.
3. Unm(x1,...,xn) = xm, where 1 £ m £ n.
4. F(x1,...,xn) = G(H1(x1,...,xn),...,Hm(x1,...,xn)), where
G,H1,...,Hm have been previously introduced.
4. F(0) = k*, F(S(x)) = G(x,F(x)), where k ≥ 0 and G has been
previously introduced.
5. F(x1,...,xn,0) = G(x1,...,xn), F(x1,...,xn,S(y)) =
H(x1,...,xn,y,F(x1,...,xn,y)), where G,H have been previously
introduced.
We present three versions of PRA.
1. 15 plus the axiom scheme
(A[x/0] Ÿ ("x)(A Æ A[x/S(x)])) Æ A
where A is quantifier free, in the context of first order
predicate calculus for our language.
2. 15 plus the rule
from A[x/0] and A Æ A[x/S(x)], derive A
where A is quantifier free, in the context of first order
predicate calculus for our language.
3. 15 plus the rule
from A[x/0] and A Æ A[x/S(x)], derive A
where A is quantifier free, in the context of free variable
predicate calculus for our language.
In the case of 3, all theorems are quantifier free (of
course, interpreted universally).
THEOREM 6.1. All three versions of PRA prove the same
sentences without quantifiers. The P2 sentences provable in
1,2 and the P2 sentences that are provable from the theorems
of 3 in predicate calculus are the same (even if blocks of 42
like quantifiers are used in the P2 sentences). This is
provable in IS0(exp).
THEOREM 6.2. Under any adequate formalization of provability
in PRA, we have that PRA proves the consistency of every
particular finite fragment of PRA. In fact, the 1consistency
(every provable S1 sentence is true). PRA proves the 1consistency of IS0(exp) + superexp. In particular, PRA is not
interpretable in IS0(exp) + superexp, or in any of its finite
fragments. This is provable in IS0(exp).
THEOREM 6.3. If PRA proves a sentence ("n)($m)(A), A
quantifier free, then there is a function symbol F such that
PRA proves ("n)(A[m/F(n)]). This is provable in IS0(exp).
A streamlined version of the Ackermann hierarchy is defined
as follows. We define functions Ak:Z+ Æ Z+ as follows. A1(n)
= 2n, Ak+1(n) = AkAk...Ak(1), where there are n Ak's.
More formally, A(1,n) = 2n, A(k+1,n+1) = A(k,A(k+1,n)).
We can ask whether each of the functions Ak exist in PRA. How
is this formalized? One way is to see that for any k ≥ 1,
there is a Turing machine algorithm corresponding to Ak, which
is constructed inductively on k by a primitive recursion. We
can then state "Ak exists" as
for all n, the algorithm
corresponding to Ak halts at input n.
We can also formalize "the Ackermann function exists" in an
analogous way, that the algorithm corresponding to our
definition of A halts at any two arguments.
THEOREM 6.4. PRA does not prove "for all k ≥ 1, the function
Ak exists". PRA does not prove "the function A exists". The
equivalence between these two statements is provable is
IS0(exp). In fact, over IS0(exp), these two statements are
provably equivalent to the 1consistency of PRA.
We can alternatively present PRA with the language L0 at the
base, using Q, while we still introduce new symbols for
primitive recursive functions in the same way. If we do this,
then we get the same systems in the usual sense, and also we
can use the least number principle instead of induction. 43
7. IS1.
IS1 is in the language L0 with 0,S,+,•,=,£. IS1 is Q together
with the induction scheme for S1 formulas. Obviously IS1
contains IS0, and hence the development of the graph of
exponentiation and finite sequence coding where there are
exponentials. But then using S1 induction, we can obviously
prove that the exponential graph is total. These ideas create
a standard interpretation of IS0(exp) in IS1. This
interpretation preserves the domain, and 0,S,+,•,=,£.
We can also view PRA as a
the algorithms associated
function can be proved to
within IS1. This fact can subsystem of IS1. This is because
with every primitive recursive
be total (halt at all inputs)
be proved in IS0(exp). As remarked before, we can view the language of PRA as
including L0.
THEOREM 7.1. IS1 and PRA prove the same P2 sentences. In
particular, a Turing machine can be proved in IS1 to halt
everywhere iff it can be proved in PRA to halt everywhere.
This fact cannot be proved in IS0(exp). Moreover, this fact is
provably equivalent to superexp over IS0(exp).
THEOREM 7.2. IS1 remains logically equivalent if we make any
of the following changes in formulation. Using order
induction for S1 formulas. Using least number principle for S1
formulas. Using order induction for P1 formulas. Using least
number principle for P1 formulas.
THEOREM 7.3. IS1 is finitely axiomatizable. The result of
applying bounded quantification to a S1 formula is provably
equivalent to a S1 formula over IS1.
8. ISn, n ≥ 2.
ISn is Q together with induction for all Sn formulas.
THEOREM 8.1. ISn remains logically equivalent if we make any
of the following changes in formulation. Using order
induction for Sn formulas. Using least number principle for Sn
formulas. Using order induction for Pn formulas. Using least
number principle for Pn formulas. 44
THEOREM 8.2. ISn is finitely axiomatizable. The result of
applying bounded quantification to a Sn formula is provably
equivalent to a Sn formula over ISn.
THEOREM 8.3. ISn+1 proves the 1consistency of ISn. In fact,
ISn+1 proves "every Sn+1 sentence provable in ISn is true".
Theorem 8.3 needs a discussion of the development of truth
predicates in ISn+1, although there is an obvious version of
it without this.
9. PA.
PA is Peano arithmetic, which is the union of the ISn.
THEOREM 9.1. PA is not finitely axiomatizable. For all n,m ≥
0, PA proves "every Sm sentence provable in ISn is true".
We end the lecture with a discussion of provably recursive
functions and Œ0, <Œ0 recursive functions, and Œ0 recursive
functions. The provably recursive functions are exactly the
<Œ0 recursive functions. See
H. Friedman and M. Sheard, Elementary descent recursion and
proof theory, Annals of Pure and Applied Logic 71 (1995), pp.
145. PHILOSOPHY 536
PHILOSOPHY OF MATHEMATICS
LECTURE 5
11/6/02
11/7/02
As promised, we begin with our preferred independence result
from PA (Peano Arithmetic). The original one of a serious
mathematical flavor was that of Paris and Harrington in 1977,
published as an Appendix in the Barwise handbook for
mathematical logic, Springer.
Arguably more mathematically natural ones were published in
my 1998 Annals of Mathematics paper, which is mostly
concerned with other matters. Also the one of Kanamori and
McAloon is arguably more natural, and an account of it has
appeared in Dave Marker's recent model theory book, Springer. 45
A number of quite different arguably more natural ones
related to the Kruskal tree theorem has appeared in my
article "Internal tree embeddings" in the Feferfest volume,
published by the ASL, 2002.
Let S(A) be the set of all subsets of A and Sk(A) be the set
of all subsets of A of cardinality k. Write A for the
cardinality of A. Let [n] = {1,...,n}.
THEOREM 1. For all k ≥ 1 there exists n ≥ 1 such that the
following holds. For every F:S[n] Æ [n] there exists E Œ
Sk[n] such that F[S(E)] « [min(E)+k] £ k.
THEOREM 2. Theorem 1 cannot be proved in PA. It is provably
equivalent to the 1consistency of PA over EFA. The growth
rate of the least n as a function of k is an Œ0recursive
function that eventually dominates every < Œ0recursive
function.
We now discuss subsystems of second order arithmetic. We
follow the notation and treatment of Simpson's authoritative
book, Subsystems of Second Order Arithmetic, Springer, 1999.
1. Z2.
Z2 is so called second order arithmetic, but it is not a
second order system. It is a two sorted first order system.
Lower case letters range over w = {0,1,2,...}. Upper case
letters range over the subsets of w. The former are the
numerical variables, and the latter are the set variables.
The numerical terms are built up from the numerical
variables, the constant symbols 0,1, and the binary function
symbols +,•. The atomic formulas are of the forms
t1 = t2
t1 < t2
t1 Œ X
where t1,t2 are numerical terms and X is a set variable.
Formulas are built up from atomic formulas by means of the
connectives Ÿ,⁄,ÿ,Æ, ´, the number quantifiers "n, $n, and
the set quantifiers "X, $X. A sentence is a formula with no
free variables. L2 is this language of second order
arithmetic. 46 There is a standard Hilbert style logically complete system
appropriate for L2.
The intended model for L2 is (w,S(w),+,•,0,1,<). An wmodel
is any model for L2 of the form (w,E,+,•,0,1,<), E Õ S(w). By
usual conventions it is required that E be nonempty.
We now present the system Z2. It is understood that we have
the usual axioms and rules of logic for L2.
1. Numerical axioms.
ÿn+1 = 0
m+1 = n+1 Æ m = n
m+0 = m
m+(n+1) = (m+n)+1
m•0 = 0
m•(n+1) = (m•n)+m
ÿm < 0
m < n+1 ´ (m < n ⁄ m = n)
2. Induction axiom.
(0 Œ X Ÿ ("n)(n Œ X Æ n+1 Œ X)) Æ n Œ X.
3. Comprehension axioms.
($X)("n)(n Œ X ´ j), where j is a formula of L2 in which X
is not free.
Z2 is very powerful. However, Z2 is not the most powerful
system one can naturally write down of a "logical" character
that is in the language L2. There is the axiom of countable
choice, and the stronger axiom of dependent choice.
("n)($X)(j) Æ ($Y)("n)($X)(j Ÿ Y = Xn)
("X)($Y)(j) Æ ($W)("n)($X)($Y)(j Ÿ X = Wn Ÿ Y = Wn+1)
where the displayed equations are suitable abbreviations.
One can attempt to prove that in some sense dependent choice
is the strongest "simple" principle in L2. There are a lot of
difficulties in carrying out such a program, but let me
mention something that seems particularly promising.
Note how simple the comprehension axiom scheme is. One can
define the general notion of a scheme. Note that the
comprehension axiom scheme is a scheme with 2 quantifiers in 47
which the only nonlogical symbol is Œ (and one schematic
letter). One can attempt to analyze all such schemes, and
determine their status in various senses. In the general
notion of scheme, one uses j syntactically as a unary
relation symbol, writing
($X)("n)(n Œ X ´ j(n))
to indicate that among the variables displayed, only n is
allowed to be free when substituting formulas for the
schematic letter j. This provides a general framework for
free variable restrictions such as in the comprehension axiom
scheme.
I have done classifications like this which are much more
difficult. See "Three quantifier sentences" on the preprint
server that I use. It will appear in Fundamenta. I would try
to classify all Œ schemes in L2 with three quantifiers.
THEOREM 1.1. Z2 without induction interprets Z2.
2. Restricted comprehension.
The most obvious way to obtain important fragments of Z2 is to
restrict the formulas j in comprehension. The most
simpleminded of these restrictions is to require that j be
arithmetical; i.e., have no set quantifiers. Then we get what
is called ACA0, for arithmetic comprehension. The naught is
because, historically, ACA was considered, where induction
was formulated as a scheme involving all formulas. For a
number of good reasons, the systems with the induction axiom
rather than the induction scheme now predominate. It is known
that under very general circumstances, the scheme is stronger
than the axiom, both in terms of outright provability and in
terms of interpretation power and consistency strength.
THEOREM 2.1. Z2 is not finitely axiomatizable. ACA0 is
finitely axiomatizable.
The system ACA0 is particularly important in light of its
connection with PA. Note that PA is a subsystem of ACA0 in the
sense that every axiom of PA is a theorem of ACA0.
THEOREM 2.2. ACA0 is a conservative extension of PA for all
arithmetic sentences. This fact is equivalent to 48
"exponentiation can be indefinitely iterated" over EFA =
IS0(exp).
It is natural to consider the extension of PA in L2 that has
no comprehension but allows induction for all arithmetic
formulas of L2; i.e., free set variables. Call this PA2.
THEOREM 2.3. ACA0 is a conservative extension of PA2 for
arithmetic formulas of L2.
THEOREM 2.4. The wmodels of ACA0 are those where E Õ S(w) is
nonempty and closed under relative arithmeticity. The
arithmetic sets form the minimum wmodel of ACA0.
A P1k formula, k ≥ 1, is a formula of L2 which starts with a
universal set quantifier followed by at most k1 set
quantifiers, followed by an arithmetic formula. The first
mentioned quantifier is allowed to be omitted.
A S1k formula, k ≥ 1, is defined analogously.
P10 and S10 is identified with the arithmetic formulas.
P1kCA0 results from Z2 by restricting j in comprehension to
P1k formulas. S1kCA0 is defined analogously.
THEOREM 2.5. Let k ≥ 0. P1kCA0 and S1kCA0 are equivalent and
finitely axiomatizable. P1kCA0 does not derive P1k+1CA0. In
fact, P1k+1CA0 is not interpretable in P1kCA0, and proves the
consistency of P1kCA0. These results hold even if we replace
P1kCA0 with P1kCA0 + IND.
Here IND is the induction scheme for all formulas of L2.
THEOREM 2.6. There is no minimal wmodel of Z2. For k ≥ 1,
there is no minimum wmodel of P1kCA0. In fact, no minimal wmodel of P1kCA0.
We had published a proof that there is no minimal wmodel of
Z2 apparently without pushing the generality. Quinsey is
credited with showing the following.
THEOREM 2.7. Let S be a recursive set of L2sentences which
includes the axioms of ATR0 (a fragment of P11CA0). Then S
has no minimal wmodel. 49
All this is in Simpson's book in detail.
A bmodel is an wmodel such that every P11 formula with
parameters from the wmodel that holds in the wmodel is
true. This turns out to be an important notion.
THEOREM 2.8. For all k ≥ 0, there is a minimum bmodel of P1kCA0. The minimum bmodel of P1kCA0 is properly included in
the minimum bmodel of P1k+1CA0, and in fact has an
enumeration there.
There is another set of simpleminded restriction of Z2 of
importance. These are D1kCA0. Here comprehension takes the
form
("n)(j ´ y) Æ ($X)("n)(n Œ X ´ j)
where j is P1k and y is S1k and X is not free in j.
THEOREM 2.9. For all k ≥ 1, P1kCA0 Õ≠ D1k+1CA0 Õ≠ P1k+1CA0,
and S1kCA0 is finitely axiomatizable. The first proper
inclusion is weak in that we have equiconsistency and
conservative extension for P11 sentences. The second proper
inclusion is strong in that we have provable consistency.
THEOREM 2.10. D11CA0 has a minimum wmodel, and this is the
hyperarithmetic subsets of w. For k ≥ 2, D11CA0 has no
minimal wmodel by Quinsey's theorem.
3. Fragments of P11CA0.
It turns out
mathematics,
even further
Mathematics,
mathematics, that in the appropriate formalization of
the vast bulk is provable in P11CA0, and in fact
down. The next lecture is about Reverse
which concerns the logical analysis of
and the main systems are subsystems of P11CA0. The most commonly encountered of the proper subsystems of P11CA0 is the system ATR0, or arithmetic transfinite recursion
naught.
ATR0 replaces the comprehension axiom scheme by a single axiom
that says the following. Let X be a well ordering of w, coded
in terms of a standard pairing function on w. Let j(n,Y) be
an arithmetic formula with set and number parameters allowed.
Note that j provides what is called an arithmetic operator 50
that sends Y to {n: j(n,Y)}. We assert that we can iterate
this operation along the well ordering X starting with any Y
Õ w. This construction is given in terms of a set W, using
its cross sections Wn, n Œ w. At limits, these are combined
into a single set in a standard way, and one then continues.
ATR0 is also the weakest of the most commonly encountered
systems for which the hyperarithmetic subsets of w do not
form an wmodel.
THEOREM 3.1. ATR0 is finitely axiomatizable and is a proper
fragment of P11CA0. In fact, ATR0 is provably consistent
within P11CA0.
We proved the following.
THEOREM 3.2. ATR0 is equiconsistent with the Feferman Schutte
analysis of predicativity in terms of the proof theoretic
ordinal G0. ATR0 is a conservative extension of the versions
of predicativity that are in L2, for all P11 sentences. This
is true even for the latter systems with IND.
The strongest of the most commonly encountered proper
subsystems of ATR0 is ACA0 which we have discussed earlier.
THEOREM 3.3. ATR0 proves the consistency of ACA0. This is true
even for ACA = ACA0 + IND. ATR0 proves D11CA0.
We now come to fragments of ACA0. Two very important ones have
emerged, RCA0 and WKL0. RCA0 is based on a comprehension axiom
scheme, but the situation is more delicate. WKL0 is an
extension of RCA0 by an important principle that is not like
comprehension at all.
RCA0 is recursive comprehension axiom naught. We need to
define the bounded arithmetic formulas of L2. We have
encountered this in the previous lecture. We will do this
using terms:
("n < t)(j) = ("n)(n < t Æ j)
($n < t)(j) = ($n)(n < t Ÿ j)
where j is a formula of L2, t is a numerical term, and n does
not appear in t. 51
The bounded arithmetic formulas are the arithmetic formulas
all of whose quantifiers are so bounded.
The S01 formulas are the arithmetic formulas of L2 which begin
with an existential number quantifier followed by a bounded
arithmetic formula. Analogously for P01.
The axioms of RCA0 consist of
1. Numerical axioms. As in Z2.
2. S01 induction.
(j[n/0] Ÿ ("n)(j Æ j[n/n+1])) Æ j, where j is S01.
3. D01 comprehension.
("n)(j ´ y) Æ ($X)("n)(n Œ X ´ j), where j is S01, y is
P01, and X is not free in j.
THEOREM 3.4. RCA0 is finitely axiomatized. RCA0 has the
minimum wmodel consisting of the recursive subsets of w.
RCA0 is a proper subsystem of ACA0. ACA0 proves the
consistency of RCA0 but not of RCA = RCA0 + IND.
THEOREM 3.5. RCA0 is a conservative extension of IS1 for all
arithmetic sentences, which is in turn a conservative
extension of PRA for P02 sentences.
We now come to WKL0. This is RCA0 augmented with "weak Konig's
lemma". This asserts that any infinite tree of finite
sequences of 0's and 1's has an infinite path. Using standard
coding mechanisms, this is easily stated in RCA0.
THEROEM 3.6. WKL0 is a subsystem of ACA0, which proves its
consistency. WKL0 has no minimal wmodel. WKL0 has an wmodel
consisting of some of the subsets of w recursive in 0' (i.e.,
D02). WKL0 is conservative over RCA0 for P11 sentences.
4. TI.
TI consists of ACA0 together with the axiom scheme
WF(X) Æ TI(X;j)
where WF(X) means that X codes a well ordering of w, and
TI(X;j) is transfinite induction on X with respect to the
arbitrary formula j. 52 THEOREM 4.1. For all k ≥ 1, TI is not provable in P1kCA0.
P11CA0 proves the consistency of TI, and even the existence
of a b model of TI. TI is not finitely axiomatizable.
THEOREM 4.2. TI proves ATR0, and the existence of an wmodel
of ATR0. TI has no minimal b model. Neither does ATR0.
5. Proof theoretic measures of systems.
Two common measures of systems are the provably recursive
functions and the proof theoretic ordinal.
Let T be a theory in L2 that contains EFA. A provably
recursive function is a recursive function f:w Æ w such that
for some index e of a Turing machine,
i) T proves ("n)({e}(n) halts);
ii) for all n, f(n) = {e}(n).
Suitable classifications of the provably recursive functions
of subsystems of Z2 has been a preoccupation of proof
theorists for many decades.
RCA0 Primitive recursive functions.
WKL0 Primitive recursive functions.
ACA0 <Œ0 recursive functions.
ATR0 <G0 recursive functions.
P11CA0 <qW_w(0) recursive functions.
TI <qW^w(0) recursive functions.
The provable ordinal of T is the sup of all ordinals a such
that for some index e of a Turing machine,
i) T proves "e defines a well ordering of w";
ii) the well ordering of w that e defines has ordinal a.
RCA0 ww.
WKL0 ww.
ACA0 Œ0.
ATR0 G0.
P11CA0 qW_w(0).
TI qW^w(0).
PHILOSOPHY 536
PHILOSOPHY OF MATHEMATICS
LECTURE 6 53
Reverse Mathematics
11/13/02
11/17/2
1. The reverse mathematics program.
Here are the five main systems of reverse mathematics.
RCA0
1. Numerical axioms.
2. S01 induction.
3. D01 comprehension.
WKL0
1.
2.
3.
4. Numerical axioms.
S01 induction.
D01 comprehension.
Weak Konig's Lemma.
ACA0 1. Numerical axioms.
2. Induction axiom.
3. Comprehension axioms. For arithmetic formulas only. ATR0
1. Numerical axioms.
2. Induction axiom.
3. All arithmetic transfinite recursions along all well
orderings of w.
P11CA0
1. Numerical axioms.
2. Induction axiom.
3. Comprehension axioms. For P11 formulas only.
These systems, as well as some others, are used to classify
the logical structure of a substantial body of mathematics. 54
One strives for the following kind of result. Let T be a
mathematical theorem. It is essential that there be a
faithful formalization of T as a sentence in the language of
Z2, which is the same as the two sorted language of these five
systems, based on 0,1,+,•,<,Œ.
There are some fundamental issues regarding such faithful
formalizations  the so called coding issues. It is important
to have a small set of primitives for many reasons, but this
comes with a cost. Obviously mathematics proceeds by building
up very substantial layers of definitions, one on top of
another, and this is not done with logical investigations in
mind.
Often mathematics proceeds with notions for which standard
unraveling into mathematical primitives involve implicit
epistemic/ontological commitments that are sufficiently
strong as to overwhelm any inherent logical structure we seek
to uncover.
In practice, we seek to develop coding mechanisms, whereby
complicated objects are reduced to the primitives of Z2 in
ways that do not destroy what we seek to uncover. This
requires care.
In the present development of RM = reverse mathematics, a set
of established coding mechanisms have emerged, which in many
cases appear unassailable. However, in other cases they need
to be justified in ways that they have not been.
I have been trying to develop some ideas about the
justifications of coding. I haven't had the time to work out
a general framework for this, but in section 3 I will present
some justifications on a somewhat ad hoc basis of the
principal coding mechanisms of RM.
Now let's come back to the main theme of RM. We start with a
mathematical theorem T, the more fundamental the better. We
assume that we have a faithful formalization of T in the
language of Z2.
We seek to determine the "logical status" of T. The preferred
way is to show that one of the logically fundamental finitely
axiomatized fragments of Z2 such as WKL0, ACA0, ATR0, P11CA0,
is outright provably equivalent to T over RCA0. That is why we
call RCA0 the base theory for RM. 55 Why is this RM program illuminating? We mention two factors.
One is the robustness of the formalization of the relevant
mathematics in the language of Z2. The other is the number of
equivalence classes.
What equivalence relation? The equivalence relation
RCA0 proves S ´ T
where S,T are appropriately formalized mathematical theorems.
Such equivalence classes are generally named by the most
logically fundamental formal system that has arisen which is
provably equivalent, over RCA0, to the elements of the
equivalence class.
As I said last week, the current bible of RM is Simpson's
book, Subsystems of Second Order Arithmetic, Springer, 1999.
In there, I would venture to say that between 10 and 15
equivalence classes are represented.
It is true that there is a special situation with Kruskal's
theorem, where there are numerical parameters which, when
varied, lead to inequivalent statements over RCA0. So this is
a way of generating a lot of equivalence classes. But this is
highly unusual and can be isolated from the discussion as
involving weakenings of a single theorem.
We now move on to the issue of linearity. The five main
systems of RM are linearly ordered under derivability.
However, there are several good examples of mathematical
theorems classified under RM where neither is provable from
the other over RCA0.
In all known cases arising out of ordinary mathematics, we
have comparability in the weaker sense. Either RCA0 + S
interprets RCA0 + T or vice versa.
In fact, we have observed a strong dichotomy. Either RCA0 + S
and RCA0 + T are mutually interpretable, or one of these two
proves the consistency of the other. We conjecture that this
kind of dichotomy will continue to be the case in the
development of RM. 56
A great deal of mathematics, when appropriately formalized in
the language of Z2, is provable in RCA0. Therefore it is not
subject to an RM analysis (except to the extent of proving it
in RCA0).
Therefore it is of great interest to weaken the base theory
of RM from RCA0 to something weaker, or much weaker.
If the base theory is dropped too far, then we may lose the
robustness of the formalizations, where a given mathematical
theorem may have a myriad of slightly different
formalizations, no two of which are provably equivalent over
the too weak base theory. This situation could be saved if
some guiding principle about formalizations proved unifying
and effective.
Research on weakening the base theory somewhat has already
begun by Simpson's work on RCA0*. Here S01 induction is
dropped in favor of induction for bounded formulas, where
exponentiation is added. We have conservation over EFA =
exponential function arithmetic = IS0(exp). The results are
somewhat encouraging, but it is still too early to tell how
good an idea this really is, and how more radical weakenings
will fare.
There is the objection that the axioms of RCA0, and even
RCA0*, are of a logical nature, and should be replaced by
purely mathematical principles. In fact, can one build up
logical power starting only from logic and direct quotes from
the fundamental mathematical literature? This would show in a
new definitive way that logical strength and the Gödel
phenomena are unremovable. They cannot be gotten around by
any wholly new way to slice the pie.
I addressed this issue with some success in a paper called
Reverse Arithmetic.
2. Some Reverse arithmetic.
The results of this section can be found in the paper Finite
Reverse Mathematics, on the mathpreprints preprint server.
We introduce the system T0, and show that it corresponds to
the system IS0 of polynomially bounded arithmetic (presented
below). 57
Let T0 be
variables
integers.
,•,<,= of
and sets. the following system in the two sorted language with
over integers and variables over finite sets of
For the integer sort, we use the language 0,1,+,linearly ordered rings. We use Œ between integers
Equality is used only between integers. The axioms of T0 are:
1. Linearly ordered ring axioms.
2. Finite interval. ($A)("x)(x Œ A ´ (y < x Ÿ x < z)).
3. Boolean difference. ($C)("x)(x Œ C ´ (x Œ A Ÿ ÿ(x Œ
B))).
4. Set addition. ($C)("x)(x Œ C ´ ($y)($z)(y Œ A Ÿ z Œ B Ÿ
x = y+z)).
5. Set multiplication. ($C)("x)(x Œ C ´ ($y)($z)(y Œ A Ÿ z
Œ B Ÿ x = y•z)).
6. Least element. ($x)(x Œ A) Æ ($x)(x Œ A Ÿ ÿ($y)(y Œ A Ÿ
y < x)).
THEOREM 2.1. T0 can be reaxiomatized as follows.
1. Linearly ordered ring axioms.
2. ($A)("x)(x Œ A ´ (y < x Ÿ x < z Ÿ j)), where j is a
bounded formula of T0 and A is not free in j.
3. Least element.
We now introduce the system K0 based on integers only. The
language of K0 is the same as that of T0, except no set
variables are allowed.
A bounded formula of K0 is a bounded formula of T0 that has no
set variables.
The axioms of K0 are as follows.
1. Linearly ordered ring axioms.
2. (j[x/0] Ÿ ("x ≥ 0)(j Æ j[x/x+1])) Æ (x ≥ 0 Æ j), where j
is a bounded formula of K0.
THEOREM 2.2. T0 and K0 prove the same formulas without set
variables.
THEOREM 2.3. A sentence in the language of IS0 is provable in
IS0 if and only if the result of relativizing each quantifier
to the nonnegative integers and replacing each S(t) by t+1 is
provable in K0 (or T0). 58
We now consider the system T1 whose axioms are
1. The axioms of T0.
2. Multiples. ($y)(0 < y Ÿ ("z)((0 < z Ÿ z < x) Æ ($w)(y =
z•w))).
Informally, axiom 2 asserts that for all integers x, the
positive integers 1,...,x have a common positive multiple.
This axiom can be viewed as mathematically essential since it
is an immediate consequence of having a usable discrete
factorial function.
An obvious consequence of T1 is
2’. ($y)(y ≠ 0 Ÿ ("z)((z ≠ 0 Ÿ z Œ A) Æ ($w)(y = z•w))).
Informally, 2’ asserts that the nonzero elements of any
finite set have a nonzero common multiple.
The axioms of K1 are as follows.
1.
2.
3.
4.
is Linearly ordered ring axioms.
x,y ≥ 0 Æ (x0 = 1 Ÿ xy+1) = xy•x);
(x < 0 ⁄ y < 0) Æ xy = 0;
(j[x/0] Ÿ ("x ≥ 0)(j Æ j[x/x+1])) Æ (x ≥ 0 Æ j), where j
a bounded formula in the language of K1. THEOREM 2.4. T1 and K1 prove the same formulas without set
variables and exponentiation.
THEOREM 2.5. A sentence in the language of IS0(exp) is
provable in IS0(exp) iff the result of relativizing each
quantifier to the nonnegative integers and replacing each
S(t) by t+1 is provable in K1. A sentence in the language of
IS0 is provable in IS0 + exp iff the result of relativizing
each quantifier to the nonnegative integers and replacing
each S(t) by t+1 is provable in T1.
The axioms of T1(!) are
1.
2.
3.
4. The axioms of T0.
0! = 1.
x > 0 Æ x! = x•(x1)!.
0 < x < y Æ ($z)(0 < z Ÿ z•(x!) = y!). 59
THEOREM 2.6. T1 is a subsystem of T1(!). T1 and T1(!) prove
the same formulas that do not mention !.
3. Coding mechanisms in RM.
Let us start with one of the most elemental coding mechanisms
of all: the coding mechanism for ordered pairs from N (the
set of all natural numbers, or w). The official pairing
function of [Si99] is
(m,n) = (m+n)2 + m.
The essential point about pairing is that there is an obvious
extension of RCA0 involving pairing, where every formula with
all of whose free variables are of the sorts of RCA0, is
provably equivalent to a formula with the same free variables
of RCA0.
The coding mechanism in the case of pairing provides an
interpretation of the extended theory in RCA0 which is correct
in the sense that this interpretation, when applied to any
formula all of those free variables are of the sorts of RCA0,
provides a formula in the language of RCA0 which is provably
equivalent to it. Since any coding mechanism is taken to be
the identity interpretation when applied to any formula of
RCA0, as a consequence we see that the extension of RCA0 is a
conservative extension of RCA0.
In our case, the relevant extension of RCA0 has new sorts for
ordered pairs, and a binary function symbol F taking natural
number arguments into ordered pair values, with the axiom
F(x,y) = F(z,w) Æ (x = z Ÿ y = w).
Let us call this extension of RCA0, RCA0(pair). Let j be a
formula in L(RCA0(pair)) all of whose free variables are in
L(RCA0). It is clear that j is provably equivalent to a
formula in RCA0 with the same free variables, since the only
atomic formulas involving the new sort are of the form
F(s1,s2) = F(t1,t2)
where s1,s2,t1,t2 are terms in L(RCA0). Each such atomic
subformula can be replaced by s1 = t1 Ÿ s2 = t2, and the
resulting formula will be provably equivalent in RCA(pair). 60
The resulting formula will have the same free variables and
be in L(RCA0).
Also, we can use the official pairing function of [Si99] and
get a formula in L(RCA0) with the same free variables which is
provably equivalent in RCA(pair).
We then want to add sets of ordered pairs. The crucial axiom
that makes this work right is that for every set of ordered
pairs, the forward image under any quadratic function, with
natural number coefficients, of two variables is a set. Also,
given any set of natural numbers, the inverse under any such
quadratic function is a set of ordered pairs.
With sets of ordered pairs, we can introduce functions from N
to N. The crucial axiom is the relationship between sets of
ordered pairs.
Next let us consider the extension of RCA0, RCA0(Qofld), by
the ordered field of rational numbers. We have the field
operations on the new sort for rational numbers, together
with a function symbol for the embedding of the natural
numbers into the rational numbers. There are a number of
obvious axioms including that the rational numbers are the
same as or minus of the ratio between natural numbers
(without dividing by zero). Also axioms asserting the ring
operations on the natural numbers correspond to the ring
operations on the nonnegative rational numbers.
From such axioms, one can prove that for each rational there
is a unique two natural numbers, which are appropriately
reduced, with a signed bit which corresponds to it. Also, for
each two natural numbers, appropriately reduced, there is a
unique rational number corresponding to it.
Using these ideas, one then sees that every formula in
L(RCA0(Qofld)) with free variables all from L(RCA0) is
provably equivalent in RCA0 to a formula in L(RCA0) with the
same free variables. Also the standard coding of the field of
rationals over RCA0 witnesses this fact.
We then add on sequences from Q, building on earlier
experience with sequences from N.
We now wish to add on the ordered field of real numbers. We
have the obvious algebraic axioms, with inequalities. But we 61
also have the crucial axiom that to every real number there
is an infinite sequence of rational numbers that converges to
it with 1/n convergence. Also to every infinite sequence of
rational numbers that is a 1/n Cauchy sequence, there is a
real number to which it converges.
The key point is that every sentence involving real numbers,
with free variables from earlier constructions, is provably
equivalent to a formula with the same free variables that
does not involve real numbers. The idea is to replace talk
about real numbers with talk about Cauchy sequences, provably
in the relevant theory.
We now come to a special case of continuous functions, where
things get somewhat more delicate.
Let us see what is involved in adding continuous functions
F:[0,1] Æ ¬. Aside from obvious axioms, we need the crucial
axiom in the form of the Stone Weierstraas theorem.
Specifically, that there exists an infinite sequence of
polynomials from [0,1] Æ ¬ which 1/n uniformly converges to
F. I.e., every such sequences converges to some such F, and
for every such F there exists such a sequence that converges
to F. This in turn depends on a prior development of infinite
sequences of polynomials. This also in turn depends on a
prior development of polynomials. All of these necessary
prior developments must be worked out carefully.
Note that in order to pull this off for continuous functions
from [0,1] into ¬, we needed to use a basic theorem of
mathematics, Stone Weierstraas, as an axiom. This will be
typical of how delicate coding issues will be resolved. I
have not attempted to go much farther with this program.
PHILOSOPHY 536
PHILOSOPHY OF MATHEMATICS
LECTURE 7
11/20/02
1. Zermelo set theory.
We now discuss systems ranging from Z2 to Z and ZC. Recall Z2
is the system of full second order arithmetic that we
discussed in the last two lectures. Z is Zermelo set theory
with the axiom of choice. The language of Z is Œ,=. 62
1.
2.
3.
4.
5.
6. Extensionality.
Pairing.
Union.
Separation.
Infinity.
Power set. ZC is Z together with the axiom of choice.
There is a delicate point about Z in connection with the
axiom of infinity. The usual formulation of the axiom of
infinity is
Infinity. ($x)(∅ Œ x Ÿ ("y Œ x)(y » {y} Œ x)).
Since we have full separation, we can define w as the
intersection of all such sets A. This is standard.
There is something not very robust about this. It is
perfectly sensible to take infinity in the form of the
existence of a set A such that
Infinity'. ($x)(∅ Œ x Ÿ ("y,z Œ x)(y » {z} Œ x)).
This form of the axiom of infinity has its advantages, since
the least such A is Vw, which is the set of all hereditarily
finite sets.
THEOREM 1.1. Z, or even ZC, does not prove Infinity'.
However, ZC with Infinity' is interpretable in Z. Infinity'
is provable in ZF.
Proof: Define Sn(w) for n ≥ 0, as follows. S0(w) = w, Sn+1(w)
= S(Sn). Define S•(w) = the union over n of Sn(w). Here S is
the power set operation. Note that each Sn(w) is a transitive
set. It is clear that S•(w) forms a model of ZC. It is easily
proved by induction on n ≥ 0 that for every x Œ Sn(w),
("y1,...,yn)((y1 Œ y2 Œ ... Œ yn Œ x) Æ y1 Œ w). Since this
does not hold of any superset of Vw, we see that no superset
of Vw is an element of Sn(w). Hence no superset of Vw lies in
S•(w). Therefore S•(w) does not satisfy Infinity'. It is easy
to manipulate the epsilon relation on w so that it behaves
like the epilson relation on Vw, and then preserve epsilon
with respect to sets that are not in w, in order to achieve
an interpretation of ZC + Infinity' in ZC. It is well known
that ZC is interpretable in Z via Gödel's constructible 63
universe, and how this argument can be developed over Z
instead of the usual ZF. It is well known how to prove
Infinity' in ZF, using replacement. QED
ZC is actually an extremely effective vehicle for the
foundations of mathematics, in that it is so powerful that
one has to go rather far to find mathematically natural
examples of theorems provable in ZFC but not in ZC, by any
reasonable standard of mathematically natural. We will
discuss such examples in the next lecture.
2. Fragments of ZC.
We have not previously discussed set theories, but rather
systems of first and second order arithmetic. In this
section, we provide fragments of ZC that correspond to the
levels of interpretation power represented by the previously
considered systems.
We start with the two main systems, PA and Z2. In most cases,
we list two versions, the first with a large number of
axioms, the second with a small number of axioms. Each system
and the one or two that come under it are mutually
interpretable and equiconsistent in the appropriate sense.
We will not use equality as primitive.
PA.
1.
2.
3.
4.
5.
6.
7.
8.
9. Extensionality.
Pairing.
Union.
Separation.
Power set.
Replacement.
Choice.
Foundation.
Every set is finite. 1. Pairing.
2. Separation.
Z2. 64
1.
2.
3.
4.
5.
6.
7.
8. Extensionality.
Pairing.
Union.
Separation.
Replacement.
Choice.
Foundation.
Infinity. 1. Pairing.
2. Separation.
3. Infinity.
IS0.
1.
2.
3.
4.
5.
6.
7.
8. Extensionality.
Pairing.
Union.
D0separation.
Choice.
Set foundation.
Cartesian product.
Every set is finite. 1. x » {y} exists.
IS0(exp) or EFA (exponential function arithmetic).
1. Extensionality.
2. Pairing.
3. Union.
4. D0separation.
5. Choice.
6. Set foundation.
7. Power set.
8. Every set is finite.
1. Pairing.
2. D0separation.
3. Power set.
ISn, n ≥ 1.
1.
2.
3.
4. Extensionality.
Pairing.
Union.
Snseparation. 65
5.
6.
8.
9. Choice.
Set foundation.
Power set.
Every set is finite. 1. Pairing.
2. Snseparation.
RCA0. Mutually interpretable with IS1.
ACA0. Mutually interpretable with PA.
ATR0. Simpson has a set theory mutually interpretable with
ATR0. Probably done much more neatly with a variant of KP =
Kripke Platek set theory.
P11CA0.
1. Extensionality.
2. Pairing.
3. Union.
4. S1separation.
5. Choice.
6. Set foundation.
7. Cartesian product.
8. Infinity.
1. Pairing.
2. S1separation.
3. Infinity.
P1nCA0, n ≥ 1.
1.
2.
3.
4.
5.
6.
7.
8. Extensionality.
Pairing.
Union.
Snseparation.
Choice.
Set foundation.
Cartesian product.
Infinity. 1. Pairing.
2. Snseparation.
3. Infinity. 66
3. Higher order arithmetic.
Let n ≥ 1. The vocabulary of Zn is
i) variables xim, 1 £ i £ n, and m ≥ 1, ranging over sort i,
where sort 1 is the natural number sort;
ii) 0,1,<,=,+,• all in sort 1;
iii) Œ, the usual connectives, and the usual quantifiers in
each sort;
iv) parentheses.
The atomic formulas are
s = t and s < t for numerical terms s,t;
t Œ x, where t is a numerical term and x is a variable of
sort 2;
y Œ z, where y,z are variables, and the sort of z is one
higher than the sort of y.
The formulas are built up as usual.
The axioms of Zn are
1. Usual numerical axioms.
2. Usual set induction.
3. All formulas ($x)("y)(y Œ x ´ j) in L(Zn), where j is a
formula in which x is not free in j.
THEOREM 3.1. Z proves the consistency of »nZn.
There is a fragment of Z that corresponds to »nZn. BZ =
bounded Z, has the axioms
1.
2.
3.
4.
5.
6. Extensionality.
Pairing.
Union.
Separation.
Infinity.
Power set. THEOREM 3.2. BZ and »nZn are equiconsistent. BZ is finitely
axiomatizable. »nZn is interpretable in BZ, but BZ is not
interpretable in »nZn.
We can get a close correspondence between Zn and fragments of
Z as follows. 67 Let Tn be
1.
2.
3.
4.
5.
6. Extensionality.
Pairing.
Union.
Separation.
Infinity.
The 0th through nth power set of w exist. Here the 0th power set of w is w, as given by Infinity (and
separation).
THEOREM 3.3. For all n ≥ 0, Tn and Zn+1 are mutually
interpretable. They are not finitely axiomatizable.
4. Theory of types.
The theory of types, TT, is similar to »nZn, except that we
have no axiom of infinity and do not have any arithmetic
primitives. The axiom of infinity has to be added.
The vocabulary of TT is as follows.
i) variables xnm of type n, where n,m ≥ 0;
ii) the usual connectives, and quantifiers in each type.
The atomic formulas are of the form
xŒy
where x,y are variables, and the sort of y is one more than
the sort of x.
The axioms are the comprehension axioms, which are formulas
of L(TT) of the form
($x)("y)(y Œ x ´ j)
where j is a formula of L(TT) in which x is not free.
The axiom of infinity, INF, is formulated with some
care. The standard way of doing this
"there is a nonempty set of type 2, where every element has a
proper superset that is an element". 68
THEOREM 4.1. TT + INF, »nZn, are mutually interpretable. They
are also equiconsistent with BZ.
We would like to match the TTn + INF with the Zm. Here TTn is
TT except that one uses only sorts 0,...,n.
THEOREM 4.2. For all n ≥ 2, TTn, Zn+1, Tn are mutually
interpretable and equiconsistent.
5. Some relevant mathematical independence results.
We discuss Borel diagonalization theory, as it relates to
independence from Z2 and he Z. We start with Cantor’s theorem.
THEOREM 5.1. In any infinite sequence of real numbers, some
real number is not a coordinate of the sequence.
One defines a sequence of nondegenerate closed intervals with
rational endpoints, shrinking to a point that lies off of the
sequence. One obtains:
THEOREM 5.2. There is a Borel measurable function F:¬ Æ ¬
such that for all x Œ ¬•, F(x) is not a coordinate of x.
F(x) may depend only on the (set of) coordinates of x.
THEOREM 5.3. There is no Borel measurable function F:¬• Æ ¬
obeying rng(x) = rng(y) Æ F(x) = F(y), such that for all x Œ
¬ •, F(x) is not a coordinate of x.
Or put positively,
THEOREM 5.4. Let F:¬• Æ ¬ be Borel measurable, where for all
x,y in ¬•, rng(x) = rng(y) Æ F(x) = F(y). There exists x Œ
¬ • such that F(x) is a coordinate of x.
THEOREM 5.5. Let f:¬• Æ ¬• be a Borel function such that
rng(x) = rng(y) Æ rng(f(x)) = rng(f(y)). There exists x such
that rng(f(x)) Õ rng(x).
For x,y Œ ¬•, define x ~ y iff y is a permutation of x.
THEOREM 5.6. Let f: ¬• Æ ¬• be such that x ~ y Æ f(x) ~
f(y). There exists x Œ ¬• such that F(x) is a subsequence of
x. 69
Let 2N be the usual Cantor space. Let s:2N Æ 2N be given by
s(x)(n) = x(n+1). f:K Æ K is called shift invariant iff f(x)
= f(sx).
THEOREM 5.7. Let f:K Æ K be a shift invariant Borel function.
There exists x Œ K such that f(x) = x(2).
Here x(2) = (x1,x4,x9,x16,…).
Let T be the circle group. f:T Æ T is double invariant if and
only if T(2x) = T(x).
THEOREM 5.8. There is a Borel f:T Æ T which agrees with every
double invariant g:T Æ T somewhere.
Let GRP(N) be the groups with domain N.
THEOREM 5.9. Let f:GRP(N) Æ GRP(N) be an isomorphically
invariant Borel function. There exists G such that f(G) is
embeddable in G.
All of the above Theorems 5.1  5.9 are provable in Z3 but not
in Z2. There is the question of how to treat Borel measurable
functions on complete separable metric spaces in Z2. This is
handled in terms of well founded trees. First Borel sets are
handled using well founded trees. Then one assigns Borel sets
to basic open sets, to be the inverse image of the basic open
sets. This treatment avoids any use of the axiom of choice
and other issues.
There is the question of coding and a direct third order
treatment of Borel measurable functions. One wants to develop
appropriate conservative extensions of Z2, and even of the
weak fragments of Z2 used in RM (reverse mathematics), such as
RCA0. This needs to be explored carefully.
We now give a theorem of Z that cannot be proved in BZ.
Let FG(N) be the space of all finitely generated groups whose
domain is N. This is a Borel subspace of the Baire space B(N)
of binary functions from N into N.
THEOREM 5.10. Let F:FG(N)• Æ FG(N) be an isomorphically
invariant Borel function. There exists x Œ FG(N)• such that
f(x) is embeddable in a coordinate of x. 70
Theorem 5.10 is provable in Z but not in BZ + AxC, »nZn, or
TT.
We sketch a proof of a sharp form of Theorem 5.1. There is
something exotic about it.
Let ¬* be the reals with the discrete topology. Is the
discrete topology is one of those worthless meaningless
things from the new new new new math? We shall see.
Granted, ¬* is silly, but ¬*• is not. The basic open sets in
¬*• are the Vx = {f Œ ¬*•: f extends x}, where x Œ FS(¬) =
set of all finite sequences from ¬. Obviously every open
(Borel) subset of ¬• is an open (Borel) subset of ¬*• but not
vice versa.
In any topological space, a set is called meager iff it is
contained in a countable union of nowhere dense sets;
comeager iff its complement is meager; Borel iff it is in the
least s algebra containing all open sets.
LEMMA 5.11. In any topological space, every Borel set differs
from an open set by a meager set.
Baire category for ¬•:
LEMMA 5.12. ¬• is not meager. In fact, no Vx is meager.
0,1 laws for ¬•:
LEMMA 5.13. Let A Õ ¬• be Borel and permutation invariant.
Then A is meager or comeager.
We say that F: ¬• Æ ¬ is Borel iff the inverse image of
every open subset of ¬ is a Borel subset of ¬•.
LEMMA 5.14. Every permutation invariant Borel f:¬• Æ ¬ is
constant on a comeager set.
THEOREM 5.15. Every permutation invariant Borel f:¬• Æ ¬
maps some argument to a coordinate of itself.
Obviously the use of the discrete topology to prove a
statement living in standard separable spaces is highly
unusual. Recall the standard well known 0,1 law: 71
Every permutation invariant Borel function f:¬• Æ ¬
is constant on a comeager set of full measure.
But this does not allow us to derive Theorem 5.1 because
For all c, {x Œ ¬•: c is a coordinate of x}
is meager and null. ...
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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, The Bible

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