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

[Aatu Koskensilta]

Quoting Joe Shipman <JoeShipman at aol.com>:

> Technically, I proposed that "V=L" be replaced by "V=M" where M is  
> the strongly constructible sets, which form Cohen's "minimal model"  
> if there is a standard model, but which are all of L if no standard  
> model exists. V=M is, in my opinion, a more principled axiom than  
> V=L because it strengthens the V=L notion "the only sets which exist  
> are the ones which the axioms say must exist given the ordinals" by  
> not assuming more ordinals than necessary. If there IS a standard  
> model, then Cohen's minimal model M exists as a set and satisfies  
> "V=M" so I don't see the axiom as unnatural; to me it just means  
> that all the sets that must exist are all the sets there are.

   However principle V=M might be, the axiom that ZFC has no  
omega-models is certainly neither natural nor principled. It's just  
another way of saying ZFC is not arithmetically sound, after all! (And  
as Robert Solovay notes, can't be true in any well-founded model.)

-- 
Aatu Koskensilta (aatu.koskensilta at uta.fi)

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