[FOM] reflection principle for inaccessible

Ali Enayat ali.enayat at gmail.com
Tue Feb 13 14:10:23 EST 2018

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


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