Montague 1961
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From: Marc Alcobé García on 26 Jan 2010 08:01 In Levy's Basic Set Theory it is read: "The axiom of replacement is, as we see, an axiom schema. As shown by Montague 1961, the fact that this axiom cannot be given as a single axiom of the basic language is not an accident but an inherent feature of set theory". The reference points to: Fraenkel's addition to the axioms of Zermelo. Essays on the foundations of mathematics. Bar-Hillel, Y. et al. (eds.) pp. 91-114 I do not have access to that source. Does anybody have an idea about what did Montague precisely state (and possibly prove) there?
From: FredJeffries on 27 Jan 2010 09:15 On Jan 26, 5:01 am, Marc Alcobé García <malc...(a)gmail.com> wrote: > In Levy's Basic Set Theory it is read: > > "The axiom of replacement is, as we see, an axiom schema. As shown by > Montague 1961, the fact that this axiom cannot be given as a single > axiom of the basic language is not an accident but an inherent feature > of set theory". > > The reference points to: > > Fraenkel's addition to the axioms of Zermelo. Essays on the > foundations of mathematics. Bar-Hillel, Y. et al. (eds.) pp. 91-114 > > I do not have access to that source. Does anybody have an idea about > what did Montague precisely state (and possibly prove) there? "The results reported here were stimulated by a question of Tarski: is Zermelo-Fraenkel set theory (obtained from Zermelo set theory by adding Fraenkel's Replacement Schema , or Ersetzungsaxiom) a finite extension of Zermelo set theory?" "THEOREM 3. No consistent extension without new constants of ZF is a finite extension of Zermelo set theory. (Thus, in particular, ZF, if consistent, is not a finite extension of Zermelo set theory.)"
From: Marc Alcobé García on 27 Jan 2010 12:14 On 27 ene, 15:15, FredJeffries <fredjeffr...(a)gmail.com> wrote: > On Jan 26, 5:01 am, Marc Alcobé García <malc...(a)gmail.com> wrote: > > > In Levy's Basic Set Theory it is read: > > > "The axiom of replacement is, as we see, an axiom schema. As shown by > > Montague 1961, the fact that this axiom cannot be given as a single > > axiom of the basic language is not an accident but an inherent feature > > of set theory". > > > The reference points to: > > > Fraenkel's addition to the axioms of Zermelo. Essays on the > > foundations of mathematics. Bar-Hillel, Y. et al. (eds.) pp. 91-114 > > > I do not have access to that source. Does anybody have an idea about > > what did Montague precisely state (and possibly prove) there? > > "The results reported here were stimulated by a question of Tarski: is > Zermelo-Fraenkel set theory (obtained from Zermelo set theory by > adding Fraenkel's Replacement Schema , or Ersetzungsaxiom) a finite > extension of Zermelo set theory?" > > "THEOREM 3. No consistent extension without new constants of ZF is a > finite extension of Zermelo set theory. (Thus, in particular, ZF, if > consistent, is not a finite extension of Zermelo set theory.)" In a review of Montague's 'Fraenkel's addition to the axioms of Zermelo' written by Hajnal it says that the proof of this theorem "makes essential use of the fact that ZF contains the axiom of regularity" Even with all this information I still do not see what is Levi's point. What could he understand by "inherent feature of set theory"? The fact that foundation (regularity) and replacement should be accepted as such?
From: Aatu Koskensilta on 27 Jan 2010 12:23 Marc Alcob� Garc�a <malcobe(a)gmail.com> writes: > What could he understand by "inherent feature of set theory"? Levy has in mind the observation that the fact that ZFC has infinitely many axioms is not just an accident of presentation but a result of logical necessity, there simply being no way of doing with finitely many axioms. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Marc Alcobé García on 27 Jan 2010 15:41
On 27 ene, 18:23, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Marc Alcobé García <malc...(a)gmail.com> writes: > > > What could he understand by "inherent feature of set theory"? > > Levy has in mind the observation that the fact that ZFC has infinitely > many axioms is not just an accident of presentation but a result of > logical necessity, there simply being no way of doing with finitely many > axioms. > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon man nicht sprechan kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus That makes a lot more sense than referred just to the axiom schema of replacement. |