[FOM] A. Kiselev claims to prove in ZF "There are no weakly inaccessible cardinals"

Joel I. Friedman jifriedman at ucdavis.edu
Fri Aug 19 17:05:22 EDT 2011

Yes, Kiselev's claim seems rather dubious, but may yet be true.  If 
true, it is an astonishing result, for reasons given below.  Let's 
look at some consequences.

Assume with Kiselev that ZF (deductively) yields there are no weakly 
inaccessible cardinals (w.i.c.)

Then ZF yields that there are no strongly inaccessible cardinals 
(s.i.c.)  (because every s.i.c. is a w.i.c.).

Now the s.i.c. k are in one-one correspondence with the natural 
models, <Vk, e> of second-order ZF2 (by quasi-categoricity).

Therefore, ZF yields that there are no models of ZF2.

And so, ZF2 also yields that there are no models of ZF2 (because ZF 
is a subtheory of ZF2).

This already would be an astonishing result!

But would we be on our way to violating Godel's Second Incompleteness 
Theorem?  It may seem so, but I don't think it actually is so.

We would have to prove the following:

If ZF2 is consistent, then one cannot prove within ZF2 that ZF2 is 
inconsistent (or is consistent).

But all we could prove, assuming Kiselev's claim, is the following:

If ZF2 is consistent, then one can prove within ZF2 that there are no 
models of ZF2.

(because even if there are no models of ZF2, it doesn't follow that 
ZF2 is formally inconsistent, especially given that ZF2 is a 
second-order theory and therefore doesn't satisfy the standard 
Completeness Theorem for first-order theories).

In any case, assuming that sets exist in the first place, and that 
there exists the iterated set-theoretical hierarchy (as the axioms of 
ZF are intended to guarantee), then it is natural to assume that we 
can extend this hierarchy into the s.i. realm.  For example, Tarski's 
Axiom of Strong Inaccessibility (SI), published in 1938, captures 
this natural intuition of classical set theory, which may be 
formulated as follows::

SI:    For every s.i.c. k1, there exists a s.i.c. k2, such that k1 < 
k2 (and we may assume that there exists at least one such s.i.c).

I've always regarded SI as intuitively consistent with ZF.  But 
according to Kiselev's claim, it follows that SI is inconsistent with 
ZF.  If this is true, it is truly astonishing.  It would deeply 
violate our classical set-theoretical intuitions, and indeed falsify 
our theories of large cardinals.  This issue has got to be resolved!

Joel Friedman

On 8/16/2011 4:45 PM, Martin Davis wrote:
>Alexander Kiselev has asked that links to this work be posted on 
>FOM. They are as follows:
>   <http://arxiv.org/abs/1010.1956>http://arxiv.org/abs/1010.1956
>   <http://arxiv.org/abs/1011.1447>http://arxiv.org/abs/1011.1447.
>Readers should be warned that this claim is regarded as very dubious.
>Martin Davis
>Professor Emeritus, Courant-NYU
>Visiting Scholar, UC Berkeley
>eipye + 1 = 0
>FOM mailing list
><mailto:FOM at cs.nyu.edu>FOM at cs.nyu.edu

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