FOM: Models of ZFC

Joe Shipman shipman at
Tue Jan 19 10:59:06 EST 1999

Struckmann writes:
>If k is a (strongly) inaccessible cardinal then it is
>well known that V_k is a model of ZFC.

>Let's look at some form of converse of this statement, namely:
>If a set u is a model of ZFC then there exists a (strongly)
>inaccessible cardinal k such that u=V_k.

>I suppose this statement to be true. Is there a reference
>and a detailed proof?

This is clearly false as stated because if ZFC has any standard models
(models where the relation is the actual membership relation for a set)
then it has countable ones.   Struckmann may have meant to ask "If V_k
is a model of ZFC must k be a strongly inaccessible cardinal?"  If
instead of ZFC you look at the set of true sentences of set theory
(which of course can't be defined in ZFC but you can use a higher theory
of sets and classes) you get a more interesting question.  Is there a
V_alpha which models exactly the true sentences of set theory?  I
propose that such a perfect reflector be called an "infallible
cardinal", because appeals to it always give the true answer.  If such a
cardinal exists, how high up must it be?
-- Joe Shipman

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