Harvey Friedman friedman at math.ohio-state.edu
Tue Oct 28 03:00:13 EST 1997

```>It seems to me that making sense
>of them [large cardinals] is THE most important and troublesome
>foundational problem of the
>day. After all, by MRDP we know that these assumptions have consequences
>asserting the non solvability of specific Diophantine equations. This
>connection of the very large with very ordinary mathematics suggests that
>(amending Frege) "the role of the [very large] infinite in arithmetic is not
>to be denied".

The following foundational problems are on my short list of problems in the
foundations of mathematics, as well as other problems:

1. Where do the axioms of set theory come from, including the large
cardinal axioms? Can they be generated by a few philosophically fundamental
principles, at least in terms of interpretability? In particular, one must
account for the impredicative separation axioms, which appear to have a
similar circularity to the inconsistent comprehension axioms.

2. Does anything like the full power of the axioms of set theory have any
necessary uses for concrete problems?

As you well know, MRDP has as a consequence the existence of Diophantine
problems for which the solution necessarily uses any level of set theory
that you want - including large cardinals. However, this theoretical result
is very unsatisfactory from a certain standpoint.

Specifically, one is unable to use MRDP to write down a specific
Diophantine problem with this metamathematical property. In partcular I
raise the, as far I know, open question:

QUESTION: Is there a polynomial whose total representation in base 10 has
at most 1000 symbols, for which the question of whether or not there is a
solution is independent of ZFC?

By total representation, I mean: the polynomial must be fully written with
coeffiencts in base 10, exponents in base 10, and subscripts on variables
in base 10.

Leo told me once that he looked into this in connection with his looking at
Jones' work, and saw that the coding is one level too high to get a result
like this.

I had decided a long time ago that what was needed for 2 was - at least -
something that meets the following criteria:

a. A sentence B in discrete mathematics which is independent of ZFC, or
even necessary uses large cardinals to prove;
b. B must be considered natural to a relevant mathematical community -
where this naturalness is evidenced by

b1. a well attended one hour talk can be given on it to this
community, and nearly everybody stays till the end
b2. the talk is generally considered interesting and successful;
b3. subsidiary questions that may or may not be independent, but
are closely related, are actually taken up by that community in an effort
to prove them in normal mathematics;
b4. there is a real area of problems that are considered
intriguing, and arguably connected with ongoing concerns.

Stay tuned.

>In the light of the above, I was particularly struck by Harvey Friedman's
>offer to show how large cardinal axioms can appear as part of a natural
>process of proceeding from the finite to the infinite. That sounds really
>interesting.

You might want to have a look at "Transfer Principles in Set Theory." which
is in Postscript form on the web at www.math.ohio-state.edu/~friedman/

Also, I have been working up some reaxiomatizations of set theory based on
three fundamental set theoretic principles: a) extensionality b) variants
of Zermelo's separation c) variants of Russell's reducibility. When done in
certain simple ways, these generate ZF and various large cardinals. They
are based on the usual language of set theory with a constant symbol W for
a subworld. Here I am taking the liberty of viewing Rusell's reducibility
in the general form that says "if there exists something somewhere with
some property then there exists something more down to earth with this (and
related) property (properties)." This resembles reflection, of course, but
is less technical and conceptually cleaner.

***COPY OF PRIVATE E-MAIL OF 10/20/97***

1. ZF\Foundation

This has been very stable, and we repeat the axiomatization. It's in the
language L(epsilon,W), with W as a constant symbol.

1. Extensionality. (forall x)(x in y iff x in z) implies (forall x)(y in x
iff z in x).
2. Subworld Separation. x in W implies (therexists y in W)(forall z)(z in y
iff (z in x & phi)), where phi is in L(epsilon,W) and y is not free in phi.
3. Reducibility. x1,...,xk in W implies [(therexists y)(phi) iff
(therexists y in W)(phi)], where phi is in L(epsilon) and has at most the
free variables x1,...,xk,y.

This system, K(W), outright proves ZF(rfn)\Foundation, formulated with
reflection. Also, every sentence provable in K(W) without W is provable in
ZF(rfn)\Foundation.

2. Indescriable and subtle cardinals.

K1(W) is in the language L(epsilon,W) as before.

1. Extensionality. Same.
2. Subworld Separation. Same.
3. Reducibility'. (x containedin W & phi) implies (therexists y,z in W)(y
containedin x & phi[x/y]), where phi is a formula in L(epsilon) whose free
variables are among x,z, and y does not occur in phi.

Here phi[x/y] means replace all free occurrences of x in phi by y.

K1(W) proves the existence of a standard model of ZF + a extremely
indescriable cardinal. ZFC + a subtle cardinal proves the existence of a
standard model of K1(W) in which W is also interpreted as a rank.

3. Measurable cardinals.

K2(W) is the following theory in L(epsilon,W) as usual. We use containedin*
to represent core inclusion; i.e., x containedin* y if and only if x is a
subset of a transitive subset of W. I.e., every element of x, every element
of element of x, etcetera, lies in W. This has some relevant philosophical
meaning.

One can, alternatively, take containedin* as primitive, with an obvious
axiom. That would work too, but we haven't pursued it fully.

1. Extensionality. Same.
2. Subworld Separation'. (therexists x in W)(forall y in W)(y in x iff (y
containedin* W & phi)), where phi is a formula in L(epsilon,W) in which x
is not free.
3. Reducibility. Same as in K(W).

K2(W) proves the existence of standard models satisfying ZF + there are
arbitrarily large measurable cardinals. ZFC + there is a nontrivial
elementary embedding from a rank into a rank proves the existence of a
standard model of K2(W). This can be somewhat weakened.

MORE NEWS:

I am happy to report that, as far as interpretability is concerned, ZF (or
ZFC) corresponds precisely to:

1. Subworld Separation.
2. Reducibility.

In other words, just remove Extensionality from K(W)!! There is a
manuscript on this, as well as all of the above. Haven't made a posting on
the web for a while, though. Too busy trying to finish stuff. Lots of
things to juggle... Let you know. More later.  HMF

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