[FOM] Independence without forcing

[Harvey Friedman]

Reply to Eisworth 8:25Pm 7/13/03.
?
When I posted Difficult? 8:45PM 7/13/03, I was, in a way, thinking of 
what Eisworth might have in mind. But now that Eisworth has focused 
his question, I can focus in on this more clearly.

>
>However, consider now a revised position:
>
>(*)
>"All statements of 'real mathematics' are decided by the axiom system ZFC +
>V=L + whatever large cardinals make sense in L; everything else is
>metamathematics."
>
Is this refutable?

Look at Proposition 4.2 of 172:Ordered Fields/Countable DST//PD/Large 
Cardinals. This is a statement that is very much in the spirit of 
some current model theory, asking for a countable real closed field 
with a distinguished discrete subgroup in which some simple property 
holds. I proved that this was equivalent, over ACA, to the 
consistency of the scheme of projective determinacy, and hence to the 
consistency of ZFC + {there exists n Woodin cardinals}_n. So this 
would seem to refute *).

Also, in #173:Borel/DST/PD 2:11AM 5/25/03, I presented a Borel 
statement corresponding to infinitely many Woodin cardinals, in the 
spirit of Borel diagonalization.

Also look at: simpler Borel/PD 8:58AM 5/26/03, where I use Turing 
degrees, and suggest how this can be replaced by "analysis" degrees.

Perhaps you want to say something like this?

**) All statements of 'real mathematics' are decided by the axiom 
system ZFC + V = L + the existence of a transitive model of "VB + 
there exists a nontrivial elementary embedding from V into V".

Then the kind of thing I was getting at in Difficult? 8:45PM 7/13/03, 
becomes relevant.

E.g., I doubt whether these powerful transitive models, even with V = 
L, helps to settle questions like

sin(2^2^2^2^2^2^2^2) > 0

or the infinitary form that, e.g.,

the set of all n such that sin(2^[n]) > 0 has density 1/2.

The new example in Difficult? 8:45PM 7/13/03 involves set theoretic 
definability, but is meant to be more "difficult". To make this idea 
"mathematical", we modify it as follows.

Let k,n >= 1. Look at all groups that can be presented with k 
generators and a set of words in these k generators set to the 
identity, where each word is of length at most n. Call these groups 
k,n presented.

CONJECTURE. Let k be sufficiently large. The set of all n such that 
the number of k,n presented groups up to isomorphism is even, is of 
density 1/2.

Is this conjecture a candidate for a counterexample to **)?

>Where is this leading? I am consistently amazed and thrilled at the ways
>which we (the broad community of mathematical logic) have shown that the
>answer to the question "Does mathematics need more axioms?" is a resounding
>"YES". (I may have never heard of Boolean relation theory before this last
>week, but it has been a real pleasure to read Harvey's papers over the last
>few days.)

I should mention the four person article

Does Mathematics Need New Axioms?, by Solomon Feferman, Harvey M. 
Friedman, Penelope Maddy, and John R. Steel, Bulletin of Symbolic 
Logic, Volume 6, Number 4, Dec. 2000, p. 401-446.

>
>What about the question "Will mathematics always need more axioms?"  Will
>there come a time when mathematicians agree on the truth of enough axioms so
>that we can really say "all non-metamathematical statements are decided by
>our axioms", or is there some super-incompleteness phenomenon that says "no
>matter how we choose to formalize our axioms and the definition of
>'non-metamathematical', there will always be a non-metamathematical
>statement not decided by our axioms"?

It seems extremely unlikely that with any confidence one will ever be 
able to assert that "all non-metamathematical statements are decided 
by our axioms". Because there will undoubtedly remain a pile of 
(attractive) unsolved mathematical problems.

I expect to see a nice statement in Boolean relation theory and in 
other contexts which is independent of

ZFC + V = L + the existence of a transitive model of "VB + there 
exists a nontrivial elementary embedding from V into V"

assuming this system is appropriately consistent.

So these nice statements just sit there as independence results from 
the system **) above. The status of **) is now regarded as unknown 
among set theorists, and perhaps may stay that way.

Your idea is to suggest that maybe this status might be considered 
resolved some day, as well as various things like it, and there may 
not be any reasonable idea that goes beyond such ideas in any 
significant way.

I don't know any "superincompleteness" that rules this out for "real 
mathematics".

Harvey Friedman