Well founded non-standard models?
in [Logic]
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From: Frode Bjørdal on 21 Apr 2010 11:13 Are there, or could there be, well-founded non-standard models of ZF?
From: Frode Bjørdal on 12 May 2010 17:15 On 23 apr, 19:47, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > stevendaryl3...(a)yahoo.com (Daryl McCullough) writes: > > He's asking about the converse: Is everywell-foundedmodel standard? > > I read carelessly, as you note! The answer to the actual question asked > is: yes, at least modulo isomorphism. Could you please be more precise here? Daryl Mcullough's restatemeent is pertinent but in the context creates some confusion with me as to what your answer is. Is it that (1) there are well-founded models that are not standard? Or is it that (2) all well founded models are standard? In either case it would be intrfesting to have a brief explanation or reference. Thanks!
From: Aatu Koskensilta on 20 May 2010 05:02 Frode Bj�rdal <fbenlightenment4all(a)gmail.com> writes: > Could you please be more precise here? All well-founded models are standard, modulo isomorphism. By the Mostowski collapsing lemma, any well-founded model is isomorphic to a model of form <A, epsilon> with A transitive. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Frode Bjørdal on 30 May 2010 15:57 On 20 Mai, 11:02, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Frode Bjørdal <fbenlightenment4...(a)gmail.com> writes: > > Could you please be more precise here? > > All well-founded models are standard, modulo isomorphism. By the > Mostowski collapsing lemma, any well-founded model is isomorphic to a > model of form <A, epsilon> with A transitive. Thanks. Does this not depend on the notion of model being involved? E.g., could one not in some weak theory have a model of ZF in the sense that it has a set of the Gödel numbers which code a theorem of ZF, while this set of Gödel numbers for ZF is still a well-founded set of standard natural numbers?
From: Aatu Koskensilta on 1 Jun 2010 15:50
Frode Bj�rdal <fbenlightenment4all(a)gmail.com> writes: > Does this not depend on the notion of model being involved? E.g., > could one not in some weak theory have a model of ZF in the sense that > it has a set of the G�del numbers which code a theorem of ZF, while > this set of G�del numbers for ZF is still a well-founded set of > standard natural numbers? I'm afraid I don't understand your question. What does it mean for a theory to have a set of G�del numbers? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |