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Re: arithmetic in ZF Re: arithmetic in ZF Source: http://sci.tech−archive.net/Archive/sci.logic/2005−04/msg00083.html • From: "george" <greeneg@xxxxxxxxxx> • Date: 7 Apr 2005 08:02:02 −0700 H. Enderton wrote: > In a recent post, Bhup pointed out (correctly) that we can interpret > PA into ZF. But then the post went on to say: > > >6. Hence, if ZF is consistent, then every Arithmetical proposition is > >syntactically (i.e., independently of the semantic connotations of the > >above definitions of "truth" and "satisfiability") decidable from the > >axioms of ZF. > > As George Greene remarked, this is wrong. The arithmetic sentences > provable in ZF (i.e., the arithmetic sentences whose interpretations > are provable in ZF) make up "merely" an r.e. set of sentences. One thing that misled me in that presentation was multiple ways of "interpreting" the act of "interpretation". I was thinking purely in terms of the Tarskian sense of an interpretation as a mapping from the language (or its parts) to some model−theoretic structure. But there is another available and relevant sense of "interpretation" that does NOT involve interpretations−of−a−theory−into−structures/models at all, but rather is a purely LINGUISTIC sense of "interpretation" in which one takes elements of the language of Arithmetic and simply TRANSLATES them into language−of−ZF. In particular, if one maps the interpretation to the *sentence*, it becomes ambiguous which of these two "interpretations" (of "interpretation") is meant. If you take some trivial arithmetical sentence like s(0)+s(s(0))=s(s(s(0))) (which happens to be a theorem), you could use ZF to construct a model of PA and ask what the interpretation of this sentence was in that model (0 will be the empty set, s(.) will be some unary−function−represented−as−a−ZF−set, +(.,.) will be a binary function, = will be a binary predicate, and the predicate will assign T as the truth−value of the overall sentence). BUt you could instead/also Re: arithmetic in ZF 1 Re: arithmetic in ZF just ELIMINATE everything occurring in this arithmetical language, including =, by translating it into something involving only e (there are formulations of ZF where e is the only predicate and where no function− symbols are needed at all, although the canonical dialect has them). s(x) would be x U {x}; x=y would be Az[zex<−>zey]; and + and * would be, well, complicated, but basically they would be the least ordinal bijectible with some set constructed from the two argument sets (in the case of multiplication, that set could be, conventionally, the cartesian−product of the argument sets). My point is that if one wants to talk about an "axiomatization" of arithmetic under ZF, then new "axioms" one needs are instead DEFINITIONS of the linguistic entities from the language of arithmetic in the language of ZF. I know, definitions often ARE axioms, but my point is that axioms are usually NOT definitions. I mean definitions as conservative, as purely abbreviatory. > Let's give a name to the set: > Tzf = the set of sentences of arithmetic whose interpretation is > provable in ZF. I guess "interpretation" here must be the "linguistic" interpretation, but when I was responding to Bhup, I was thinking in terms of the model−theoretic one. > Then Tzf is a theory (it is closed under provability) and, being > r.e., it is a recursively axiomatizable theory extending PA. > (Tzf proves Con PA.) But there is a problem with thinking about Con PA in this context, because Neither PA NOR ANY OTHER PARTICULAR AXIOMATIZATION of first−order arithmetic IS EVEN RELEVANT to this particular theory! One important thing about Con(PA) in the context of G1 was that it was IN THE SAME language as PA. In this context, though, Con(PA) is in the language of ZF and there is no reason to think of "the axioms of PA" as axioms. They are just theorems of ZF, and the question of what a provability− predicate expressing provability FROM PA would even look like gets COMPLICATED. The axioms of ZF are available for use in proving things at every step in this framework. HOw are you supposed to express about a derivation that it needs the PA−theorems but NOT the other/additional axioms−of−ZF, when in fact ALL those axioms are NEEDED to prove the theorem−hood of the PA−axioms?? > And someone once told me a "natural" recursive axiomatization of > Tzf. Unfortunately, I have forgotten what it is. Does someone > out there in sci.logic land know what it is? Re: arithmetic in ZF 2 Re: arithmetic in ZF I don't, but I share your hope that somebody does. The only point I wanted to make (aside from confessing my own confusion under ambiguity of "interpretation") was that beyond the ZF axioms themselves, any axioms you even MIGHT possibly need to add are just definitions. . • Follow−Ups: ♦ Re: arithmetic in ZF ◊ From: Bhupinder Singh Anand • References: ♦ arithmetic in ZF ◊ From: H. Enderton • Prev by Date: Re: Is this a fallacy? • Next by Date: Uniform models • Previous by thread: Re: arithmetic in ZF • Next by thread: Re: arithmetic in ZF • Index(es): ♦ Date ♦ Thread Re: arithmetic in ZF 3

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set theory, post messages, transfer principle, consistency proof, peano arithmetic, first-order logic, halting problem, axiom of choice, standard model, d. nguyen, zf set theory, formal system, natural numbers, respect to, provability predicate

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posted: | 9/3/2009 |

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