[FOM] reflection principle for inaccessible

[Paul Blain Levy]

Thanks, Ali.  And thanks too to Paul Larson, who also sent me this
counterexample.

Paul

> Date: Tue, 13 Feb 2018 14:10:23 -0500
> From: Ali Enayat <ali.enayat at gmail.com>
> To: fom at cs.nyu.edu
> Subject: [FOM] reflection principle for inaccessible
> Message-ID:
> 	<CAPzKPNu5JcEMU_xWoTT_kqJ2NVi92_Y-fu4QKZLVaX8qVpY1Lg at mail.gmail.com>
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>
> This is a reply to the following question posed by Paul Blain Levy in
> his Feb 11 posting:
>
>>> Let phi be a sentence (no free variables) such that ZFC proves "For
>>> every strongly inaccessible kappa, phi relativized to kappa".
>>> Does it follow that ZFC proves "If a strong inaccessible exists, then phi"?
> I presume that in the condition "phi relativized to kappa" was meant
> to be "phi relativized to V_kappa", where V_kappa is the set
> consisting of all sets whose ordinal rank is less than kappa.  With
> this presumption, the answer to Levy's question is in the negative,
> assuming that there are at least two inaccessible cardinals.
>
> Here is the proof: consider the sentence phi that expresses ""For
> every ordinal gamma there is an ordinal alpha greater than gamma such
> that V_alpha is a model of ZFC".  It is easy to see -- using a
> Loewenheim-Skolem argument that can be readily implemented with ZFC --
> that if kappa is inaccessible, then phi holds in V_kappa.
>
> It remains to exhibit a model of ZFC in which phi is false but an
> inaccessible cardinal exists. Assuming the existence of at least two
> inaccessible cardinals, let kappa be the first inaccessible cardinal,
> and let alpha be the first ordinal above kappa such that V_alpha is a
> model of ZFC. The desired model of ZFC in which phi fails is V_alpha.
>
> Best regards,
>
> Ali Enayat