[FOM] A proof that ZFC has no any omega-models

Aatu Koskensilta Aatu.Koskensilta at uta.fi
Wed Feb 13 15:15:18 EST 2013

Quoting Monroe Eskew <meskew at math.uci.edu>:

> If alpha is the least ordinal such that L_alpha satisfies ZFC, then  
> L_alpha must think there is no transitive model of ZFC, by the  
> absoluteness of the Gödel operations.  Shoenfield's absoluteness  
> theorem does not apply to this situation because its hypothesis  
> includes that the model in question contains all countable ordinals.

   In the minimal model L_alpha there is no standard model of ZFC. But  
there is, in L_alpha, an (omega-)model of ZFC + all arithmetical  
truths. Such models are of course necessarily non-standard.

Aatu Koskensilta (aatu.koskensilta at uta.fi)

"Wovon man nicht sprechen kann, darüber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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