Unformatted text preview: RELATIONAL SYSTEM THEORY
ABSTRACT
by
Harvey M. Friedman
http://www.math.ohiostate.edu/%7Efriedman/
May 26, 2005
1. Introduction.
2. Ternary.
3. Systems, Subsystems, Reductions, Full Systems, Complete
Systems.
4. Results.
5. Schematic Versions. 1. Introduction.
Here we present some formal systems concerning ternary
relations which relate to informal conceptual ideas that
are arguably more fundamental than those that drive modern
set theory.
We establish mutual translations between these formal
systems and various systems of set theory, ranging from
countable set theory, ZFC, to large cardinal hypotheses
such as the existence of a Ramsey cardinal, and the
existence of an elementary embeddings from V into V.
The primitives are identity, and x[y,z,w]. The latter is
read “x is a ternary relation which holds at the objects
y,z,w.”
The use of x[y,z,w] rather than the more usual y Œ x has
many advantages for this work. One of them is that we have
found a convenient way to eliminate any need for axiom
schemes. All axioms considered are single sentences with
clear meaning. (In one case only, the axiom is a
conjunction of a manageable finite number of sentences).
The theories are single sorted, and are based on the idea
that some objects appear as arguments of other objects
(ternary relations), whereas some objects do not so appear.
The former objects are called arguments or argumental,
whereas the remaining objects are called nonarguments or
nonargumental. This is totally analogous to one sorted
formalizations of the theory of classes where sets are
defined to be classes that are an element of a class. The basic axioms form a system called Ternary, which are,
with one minor exception, all easy consequences of the
axiom scheme asserting the following. We can form an object
x which holds of any argumental objects y,z,w if and only
if a given first order formula holds of y,z,w, where x does
not appear and where parameters for objects are allowed,
and all quantifiers range over argumental objects only. The
minor exception is the axiom that asserts that there is an
argument without any arguments. As indicated above, here we
avoid any use of axiom schemes.
Although Ternary is not strong enough to derive the above
mentioned axiom scheme, it does derive instances sufficient
for our purposes when combined with the main axioms.
The main axioms are stated in terms of what we call 3systems. A 3system consists of three objects x,y,z, where
x and y and z all have at least one argument
(nonemptiness), and where every two distinct arguments of
any of x or y or z combined, are “related” by x or y or z,
using some third argument. We use the most appropriate
notion of subsystem of 3systems.
We use two different notions of reduction of 3systems. One
is areduction, where argumental components remain the same
and nonargumental components become argumental.
The other is nareduction, where argumental components
remain the same and nonargumental components remain
nonargumental, but are cut back.
All of the axioms considered take the following two general
forms: Every 3system (of a specific kind) has an
areduction (of the same specific kind).
Every 3system (of a specific kind) has an
nareduction (of the same specific kind).
The axioms of the first form correspond to roughly ZF and
fragments thereof. The axioms of the second form correspond
to large cardinal hypotheses ranging from roughly Ramsey
cardinals (just below measurable cardinals) and the
existence of an elementary embedding from V into V (over
NBG without choice). Now let me now go somewhat beyond what has been established
at this point.
I believe that set theory is the canonical mathematical
limit of informal common sense thinking. Let me explain
with an example you are all familiar with.
People, using common sense, think about, say, a full head
of hair. They think that if you remove one strand of hair
from a full head of hair, then it remains a full head of
hair.
Scientific thinking has a problem with this. After all, one
can perform a thought experiment whereby the number of
strands of hair is counted, and pulled out one by one,
until there are no more. Clearly complete baldness is not a
full head of hair.
At this point, set theory enters the picture. The idea of a
full head of hair is associated with the precise set
theoretic notion of: infinite set. It is provable in set
theory that if you take an element out of an infinite set,
then it remains infinite. It is provable in set theory that
infinite sets are not empty. It is provable in set theory
that infinite sets cannot be numerically counted  the
count never terminates.
There is a more sophisticated idea of this sort. There is
the common sense idea of a large system. Not just an inert
clump like a head of hair, but rather a large system with a
number of interlocking components with complicated internal
connections. Like the physical universe, or like the human
body, or like the world of living organisms, or like the
world economic system.
There is the idea that in any large system, we can take
something away without the system falling apart.
In the case of the head of hair, it doesn’t make any
difference which strand of hair you take away. Here, we are
merely asserting that something can be taken away  not
that anything can be taken away. And I am not asserting
that you can get away with taking only one item away. It
may be a significant amount of stuff. Furthermore, in any large system, we can take something
away without the system falling apart, and where the system
remains “similar”.
So what are the missing parts of this analogy?
1. Any large clump stays large after some (any) point
removal
Æ infinite sets.
2. Any large system remains a large system after some
portion removal Æ XXX.
3. Any large system remains a large system, unaffected,
after some portion removal Æ YYY.
4. Any large system remains unaffected after some expansion
Æ ZZZ.
XXX ~ Jonsson cardinals ~ Ramsey cardinals.
YYY ~ ZZZ ~ elementary embedding axioms roughly around a
rank into itself.
We have already backed up these statements with precise
theorems in set theory involving set theoretic structures
of sufficiently large cardinality. We call this subject,
the theory of large algebras. Initial developments along
these lines have been published in [Fr04]. Some further
developments are implicit in the unpublished notes [Fr03].
The break point (how large is large?) depends on the notion
of “unaffected” one uses. In [Fr04}, the break point is the
first measurable cardinal. The break points discussed in
[Fr03] are much higher. Some relevant notions are language
oriented, whereas others are more directly mathematical.
But the bold new idea here goes well beyond any theory of
large algebras within set theory. Here is what we
anticipate:
A. There is a nonmathematical common sense oriented theory
of systems and components which corresponds to various well
studied levels of set theory with/without large cardinals.
B. More generally, one can formulate transparent principles
of a plausible nature about ANY common sense notions, which
correspond to various well studied levels of set theory
with/without large cardinals. A word of caution here. It may well be the case that in any
region of common sense thinking, if one goes far enough,
one reaches outright contradictions. That is to be
expected. What we are anticipating is the designation of
formal systems closely associated with commonsense
thinking, that are in some ways extrapolations of
commonsense thinking, and in other ways are restrictions of
commonsense thinking, which correspond exactly to various
levels of abstract set theory.
As we shall see, A is arguably already implicit in the
development below, where we focus entirely on the notion of
a ternary relation. However, we look forward to reworking
the development here based on richer notions that are an
integral part of everyday thinking.
In fact, it can be argued that common sense thinking is
incredibly richer, logically, than mathematics or science.
Of course, it is not subject to the same kind of deep and
subtle constraints that mathematics and science operate
under. In a separate longstanding development, I struggle,
with increasing success, to find normal mathematics that
requires large cardinals. But I speculate that large
cardinals are everywhere implicit in common sense thinking.
We will hopefully get to the point where set theory with
large cardinals emerges as the one mathematical area which
applies to just about everything outside of science across the board.
In fact, set theory with large cardinals may be to common
sense thinking as the Newton/Leibniz calculus is to
science.
The “calculus” aspect of set theory with large cardinals is
as follows. There will be a proliferation of natural formal
systems involving various groups of common sense notions.
One will want to know how these systems compare under
interpretability. One will see that, in fact, there is a
quasi linear ordering under interpretability. One will want
to “calculate”, for any pair of such systems, how they
compare in this quasi linear ordering.
The only way to make such comparisons will be to have a
manageable set of representatives for each level that
arises, and first identify where each of the two systems to
be compared fits in. The manageable set of representatives is, of course, just
various well studied levels of set theory with large
cardinals.
So set theory with large cardinals may be the appropriate
measuring tool for the comparison of systems based on
common sense notions. Perhaps it can then emerge as the
most generally and transparently useful area of modern
mathematics. 2. Ternary.
The base theory for our investigation is the system
Ternary. Ternary is a one sorted theory, with only equality
and x[y,z,w].
We say that x is an argument (arg) if and only if
($y,z,w)(y[x,z,w] ⁄ y[z,x,w] ⁄ y[z,w,x]). We say that x is
argumental if and only if x is an argument. We say that x
is nonargumental if and only if x is not an argument.
Here are the axioms of Ternary.
ATOM. ($arg x)("y,z,w)(ÿx[y,z,w]).
COMPLEMENTATION. ($x)("args y,z,w)(x[y,z,w] ´ ÿu[y,z,w]).
UNION. ($x)("y,z,w)(x[y,z,w] ´ (u[y,z,w] ⁄ v[y,z,w])).
ATOMIC COMPREHENSION. ($x)("args y,z,w)(x[y,z,w] ´ j),
where j is an atomic formula not mentioning x.
PROJECTION. ($x)("args y,z,w)(x[y,z,w] ´ ($u)(v[u,z,w])).
Obviously, every one of these axioms is a single sentence,
except Atomic Comprehension. But here there are only
finitely many instances up to change of variables. E.g., we
can insist that j be an atomic formula whose variables are
among y,z,w,t,u,v. 3. Systems, Subsystems, Reductions, Full Systems,
Complete Systems.
We say that x is argumental if and only if x is an
argument. We say that x is nonargumental if and only if x is not an argument. We use the term “object” for any x.
Hence objects can be argumental or nonargumental.
We say that x is a subobject of y if and only if
("t,u,v)(x[t,u,v] Æ y[t,u,v]). We write this as x Õ y. We
write x ≡ y if and only if x Õ y Ÿ y Õ x.
Note that x Õ y does not really say that x is similar to y
 at least not very strongly. We would also like the inner
workings of x to be the same as the inner workings of y,
with regard to objects that fall within the purview of x.
This motivates the following definition.
We say that x’ is a restriction of x if and only if for all
arguments t,u,v of x’, x[t,u,v] ´ x’[t,u,v].
We will not be using the above definition. Instead, we work
with triples of objects x,y,z. The components of the triple
x,y,z are x and y and z.
We say that t is an argument of the triple x,y,z if and
only if t is an argument of at least one of its three
components.
We say that two distinct object t,u are related by x if and
only if x holds of some three objects that include both
t,u.
A 3system is a triple x,y,z, where
i. x,y,z each have at least one argument.
ii. Any two distinct arguments of the triple x,y,z are
related by at least one of its three components.
We say that x’,y’,z’ is a subsystem of the 3system x,y,z
if and only if x’,y’,z’ is a 3system such that for all
arguments t,u,v of the 3system x’,y’,z’,
x[t,u,v] ´ x’[t,u,v];
y[t,u,v] ´ y’[t,u,v];
z[t,u,v] ´ z’[t,u,v].
We introduce two related notions of reduction.
An areduction of a 3system S is a subsystem of S, where
the argumental components of S remain the same, and the
nonargumental components of S become argumental. An nareduction of a 3system S is a subsystem of S, where
the argumental components of S remain the same, and the
nonargumental components of S remain nonargumental, but not
≡.
More formally, let x,y,z be a 3system. We say that
x’,y’,z’ is an areduction of the 3system x,y,z if and
only if
i. x’,y’,z’ is a subsystem of x,y,z.
ii. x argumental Æ x = x’.
iii. y argumental Æ y = y’.
iv. z argumental Æ z = z’.
v. x nonargumental Æ x’ argumental.
vi. y nonargumental Æ y’ argumental.
vii. z nonargumental Æ z’ argumental.
We say that x’,y’,z’ is an nareduction of the 3system
x,y,z if and only if
i. x’,y’,z’ is a subsystem of x,y,z.
ii. x argumental Æ x = x’.
iii. y argumental Æ y = y’.
iv. z argumental Æ z = z’.
v. x nonargumental Æ (x’ nonargumental Ÿ ÿx’ ≡ x).
vi. y nonargumental Æ (y’ nonargumental Ÿ ÿy’ ≡ y).
vii. z nonargumental Æ (z’ nonargumental Ÿ ÿz’ ≡ z).
We say that a 3system S is full if and only if every
subobject of an argument of S is an argument of S.
We say that a 3system S is complete if and only if every
argumental object agrees with some argument of S at all
triples of arguments of S.
More formally, we say that a 3system S is complete if and
only if ("args x)($y)(y is an argument of S Ÿ
("t,u,v)(t,u,v are arguments of S Æ (x[t,u,v] ´
y[t,u,v]))).
We are now prepared to state the following axioms.
AREDUCTION. Every 3system has an areduction.
NAREDUCTION. Every 3system has an nareduction. FULL AREDUCTION. Every full 3system has a full areduction.
FULL NAREDUCTION. Every full 3system has a full nareduction.
COMPLETE AREDUCTION. Every complete 3system has a
complete areduction.
COMPLETE NAREDUCTION. Every complete 3system has a
complete nareduction.
There is a weak form of AReduction that we also work with.
We say that a 3system is argumental if and only if each of
its three components is argumental.
ARGUMENTAL SUBSYSTEM. Every 3system has an argumental
subsystem. 4. Results.
THEOREM 4.1. Ternary + AReduction is mutually
interpretable with Z_2 = second order arithmetic. The same
is true of Ternary + Argumental Subsystem.
THEOREM 4.2. Ternary + NAReduction is mutually
interpretable with NBG + “On is a Ramsey cardinal”. In
particular, it interprets ZFC + “there exists an almost
Ramsey cardinal”, and is interpretable in ZF\P + “there
exists a Ramsey cardinal”. The same is true of Ternary + AReduction + Full AReduction + NAReduction.
THEOREM 4.3. Ternary + Full AReduction is mutually
interpretable with ZFC. The same is true of Ternary + AReduction + Full AReduction.
THEROEM 4.4. Ternary + Full NAReduction, Ternary +
Complete AReduction, are both inconsistent.
We say that V(k) is strongly inaccessible if and only if
every function from an element of V(k) into V(k) is itself
an element of V(k).
THEOREM 4.5. Ternary + Complete NAreduction interprets NBG
+ {there exists a nontrivial Sn elementary embedding from V
into V}n, and is interpretable in ZF + “there exists a
cardinal k and a nontrivial elementary embedding from V(k+1) into V(k+1), where V(k) is strongly inaccessible”. The same
is true of Ternary + AReduction + Full AReduction + NAReduction + Complete NAReduction.
COROLLARY 4.6. Ternary + Complete NAreduction interprets
ZFC + “there exists a nontrivial elementary embedding from
a rank into itself”.
Here a Ramsey cardinal is a cardinal k such that for all
partitions of the finite subsets of k into two pieces, there
is a subset of k of cardinality k such that any two finite
subsets of the subset of the same finite cardinality lie in
the same piece.
An almost Ramsey cardinal is an uncountable cardinal k such
that for all partitions of the finite subsets of k into two
pieces, there is a subset of k, of any given cardinality <
k, such that any two finite subsets of the subset of the
same finite cardinality lie in the same piece.
Almost Ramsey cardinals are incompatible with the axiom of
constructability.
In [Mi79], the Dodd Jensen core model is used in order to
establish the mutual interpretability of
ZFC + "there exists a Jonsson cardinal".
ZFC + "there exists a Ramsey cardinal".
For our results (Theorems 4.2, 5.3), we use the mutual
interpretabilty of the following triple and the following
pair:
NBG + “On is a Jonsson cardinal”.
NBG\P + “On is a Jonsson cardinal”.
NBG + “On is a Ramsey cardinal”.
ZF\P + "there exists a Jonsson cardinal".
ZF\P + "there exists a Ramsey cardinal".
Mitchell has confirmed that his published proof will
establish this with “tentative strong belief”.
It is well known that NBG + {there exists a nontrivial Sn
elementary embedding from V into V}n is stronger than ZFC +
many measurable cardinals. Using known inner model theory,
it is well known that it is stronger than ZFC + projective determinacy, or ZFC + Woodin cardinals. Woodin, using
forcing arguments, has shown that NBG + {there exists a
nontrivial Sn elementary embedding from V into V}n
interprets ZFC + “there exists a nontrivial elementary
embedding from a rank into itself”. Hence Corollary 4.6. 5. Schematic Versions.
We now freely use schemes. While we believe that the
avoidance of schemes is an important development, we also
believe that there is still some importance to be attached
to the systems based on schemes.
The most mild use of schemes is to use the following.
ARGUMENTAL COMPREHENSION. ($x)("args y,z,w)(x[y,z,w] ´ j),
where j is a formula not mentioning x, whose quantifiers
range over argumental objects only.
We can also use this stronger form.
COMPREHENSION. ($x)("args y,z,w)(x[y,z,w] ´ j), where j is
a formula not mentioning x.
THEOREM 5.1. All of the systems discussed In 4.1 4.6 are
equivalent to the systems obtained by replacing Ternary
with Atom + Argumental Comprehension.
If we use Atom + Comprehension instead of Ternary, then the
systems are somewhat stronger. We restate the results using
Atom + Comprehension as follows.
THEOREM 5.2. Atom + Comprehension + AReduction is mutually
interpretable with Z3 = third order arithmetic. The same is
true of Atom + Comprehension + Argumental Subsystem.
THEOREM 5.3. Atom + Comprehension + NAReduction is
mutually interpretable with ZF\P + “there exists a Ramsey
cardinal”. In particular, it interprets ZFC + “there exists
an almost Ramsey cardinal” and is interpretable in ZFC +
“there exists a Ramsey cardinal”. The same is true of Atom
+ Comprehension + AReduction + Full AReduction + NAReduction.
THEOREM 5.4. Atom + Comprehension + Full AReduction is
mutually interpretable with MKGC = Morse Kelley with Global Choice. The same is true of Atom + Comprehension + AReduction + Full AReduction.
THEROEM 5.5. Atom + Comprehension + Full NAReduction, Atom
+ Comprehension + Complete AReduction, are both
inconsistent.
THEOREM 5.6. Atom + Comprehension + Complete NAreduction
interprets MK + {there exists a nontrivial Sn elementary
embedding from V into V}n, and is interpretable in ZF +
“there exists a cardinal k and a nontrivial elementary
embedding from V(k+1) into V(k+1), where V(k) is strongly
inaccessible”. The same is true of Atom + Comprehension +
AReduction + Full AReduction + NAReduction + Complete
NAReduction.
COROLLARY 5.7. Atom + Comprehension + Complete NAreduction
interprets ZFC + “there exists a nontrivial elementary
embedding from a rank into itself”.
There are other ways in which schemes can be used for
alternative formulations. We will not go into this here.
Earlier versions of this work employed schemes in many ways
that we have avoided here.
All of our results (sections 4 and 5) remain the same if we
use extensionality. Of course, if we use extensionality,
then there is no need to use ≡. REFERENCES
[Fr03] `Restrictions and extensions', February 17, 2003, 3
pages, draft, http://www.math.ohiostate.edu/%7Efriedman/manuscripts.html
[Fr04] Working with Nonstandard Models, in: Nonstandard
Models of Arithmetic and Set Theory, American Mathematical
Society, ed. Enayat and Kossak, 7186, 2004.
[Mi79] W. Mitchell, Ramsey Cardinals and Constructibility,
JSL, Vol. 44, No. 2 (June 1979), 260266. ...
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