FOM: Models of ZFC
Stephen G Simpson
simpson at math.psu.edu
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
assertion.
> 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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