proofs of second incompleteness

[adriano paolo shaul gershom palma]

Good evening , if at all possible, would you kindly provide a complete
citation of where/when the Jech proof was done?

Thank you in advance


















*wehrlos aber nicht ehrlos*


Otto Wels
1933.3.23


On Tue, Sep 6, 2022 at 9:54 PM Richard Kimberly Heck <richard_heck at brown.edu>
wrote:

> On 9/5/22 11:59, martdowd at aol.com wrote:
>
> FOM:
>
> Does anyone know of any proofs of the second incompleteness theorem, other
> than the standard proof-theoretic proof that
>  1) if PA is consistent then "I am not provable" is not provable
>  2) this can be formalized
>
> The proof that Bezboruah and Sheparadson gave of G2 for Q is not of this
> kind. The proof is actually given for a deductively stronger system people
> now call PA- (and shows that PA- does not prove the consistency of ANY r.e.
> theory even pure logic). But this is still a weak system, interpretable in
> Q, and their proof does not work for stronger systems, so it's very much a
> special case.
>
> Thomas Jech gave a purely semantic proof of G2 for ZFC, here:
>
>     https://www.jstor.org/stable/2160398
>
> Jech describes a way of proving this for PA by using the fact tha ACA_0
> proves the completeness theorem and is a conservative extension of PA. But I'm
> curious whether there's a more direct proof using the arithmetized
> completeness theorem and some coding of sufficiently complex (but still
> arithmetical) sets. The arithmetized completeness theorem implies that
> every consistent theory has a model of pretty low complexity.
>
> Riki
>
>
> --
> ----------------------------
> Richard Kimberly (Riki) Heck
> Professor of Philosophy
> Brown University
>
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