[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
>
>
>Martin Davis
>Professor Emeritus, Courant-NYU
>Visiting Scholar, UC Berkeley
>eipye + 1 = 0
>
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