FOM: Models of ZFC

Stephen G Simpson simpson at
Tue Jan 19 10:45:29 EST 1999

Werner Struckmann writes:

 > 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.

No, this is not the case.  

One relevant result: If there exists a transitive model of ZFC, then
there exists a countable transitive model of ZFC.  This remark comes
up in Cohen's work on the continuum hypothesis.  The proof is as
follows.  Start with a transitive model M of ZFC.  By the
L"owenheim-Skolem theorem, let M_1 be a countable elementary submodel
of M.  Then by the Mostowski collapsing lemma, let M_0 be a transitive
model isomorphic to M_1.

This provides a counterexample to your statement, because a countable
transitive model is certainly not of the form V_k.

Another relevant result: If there exists an inaccessible cardinal k,
then there exists a strong limit cardinal l of cofinality omega, such
that V_l is a model of ZFC.  This can also be proved by a
L"owenheim-Skolem argument.  V_l is also a counterexample to your

 > Institut fuer Software
 > Abteilung Programmierung

Apparently you are a computer scientist.  Did your question arise from
some computer science issues?  I'm curious ....

-- Steve Simpson

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