Models of set theory vs. models of first order theories
in [Logic]
|
Prev: found the 10% in Geometry; Poincare dodecahedral space; 36 degree twist; matched-circles #378 Correcting Math
Next: The set theoretic content of the transfinite recursion theorem
From: Marc Alcobé García on 5 Feb 2010 08:45 > It seems odd unless one looks at the particular technical mechanics of it. I know it maybe is too much to ask, but could you briefly sketch these technical details? The original question was: >What is exactly V? You said: > In Z (and its offshoots such as ZF), we have a theorem: > ~ExAy(y=y -> yex). > Now, if we then use abstraction notation to write: > {y | y=y} > and define a constant 'V' by > V = {y | y=y} > then we find that the set abstraction "term" does not properly refer > and so the definition of 'V' is not proper. I am not sure what you mean with the sentence 'the set abstraction "term" does not properly refer and so the definition of 'V' is not proper' > Therefore, how one takes the term 'V' depends on ones handling of > improperly referring terms. I (and a good number of authors) prefer to > use the Fregean method, so that, TECHNICALLY, V = 0. I wonder at which level that assignment is done. > However, we also recognize that we may refer to such notation as 'V' > and {y | y=y} as either shorthand, or (by more intricate syntactical > methods, as defined terminology that reduces to > META-language about FORMULAS of Z set theory. Yes, Levy's approach is that every formula containing class terms (in a language that is a conservative extension of the basic language of set theory) is equivalent to another formula in the basic language of set theory, but I do not see how could that help in understanding the nature of V.
From: MoeBlee on 5 Feb 2010 10:43 On Feb 5, 6:56 am, Marc Alcobé García <malc...(a)gmail.com> wrote: > You said: > > > it might help to be clear where the relativization method is > > being used not to prove consistency itself, but rather RELATIVE > > consistency > > and also gave an example of statements of consistency and relative > consistency. > > My question is: when is a consistency proof using relativization not a > relative consistency proof? When the theorem proved is not a relative consistency assertion but rather a plain conistency assertion. > Can these kinds of proof be distinguished? Yes, by what the theorem proved is. If the theorem is a plain consistency statement then the proof is a plain consistency proof. If the theorem is a relative consistency statement, then the proof is relative consistency proof. MoeBlee
From: MoeBlee on 5 Feb 2010 11:02 On Feb 5, 7:45 am, Marc Alcobé García <malc...(a)gmail.com> wrote: > > It seems odd unless one looks at the particular technical mechanics of it. > > I know it maybe is too much to ask, but could you briefly sketch these > technical details? To save a lot of typing, I would much rather recommend Suppes's 'Introduction To Logic' as including the best introduction to the subject of definitions in mathematics. I would guess that a used copy would be available at a pretty cheap price on the Internet. Or, if you like, tell me what books on mathematical logic and/or set theory you have; then, if one of those books covers the subject, I'll direct you to the relevant pages. > The original question was: > > >What is exactly V? > > You said: > > > In Z (and its offshoots such as ZF), we have a theorem: > > ~ExAy(y=y -> yex). > > Now, if we then use abstraction notation to write: > > {y | y=y} > > and define a constant 'V' by > > V = {y | y=y} > > then we find that the set abstraction "term" does not properly refer > > and so the definition of 'V' is not proper. > > I am not sure what you mean with the sentence 'the set abstraction > "term" does not properly refer > and so the definition of 'V' is not proper' Set abstraction terms of the form: {y | P} where 'P' is a formula. The set abstraction term does not properly refer when ~ExAy(yex <-> P). For example (and I'm in context of ZF unless otherwise stated in this conversation): ~ExAy(yex <-> y=y). > > Therefore, how one takes the term 'V' depends on ones handling of > > improperly referring terms. I (and a good number of authors) prefer to > > use the Fregean method, so that, TECHNICALLY, V = 0. > > I wonder at which level that assignment is done. There are different variations in handling this that might occur at the object level or the meta-level depending on the details of the handling. > > However, we also recognize that we may refer to such notation as 'V' > > and {y | y=y} as either shorthand, or (by more intricate syntactical > > methods, as defined terminology that reduces to > > META-language about FORMULAS of Z set theory. > > Yes, Levy's approach is that every formula containing class terms (in > a language that is a conservative extension of the basic language of > set theory) is equivalent to another formula in the basic language of > set theory, but I do not see how could that help in understanding the > nature of V. The point was to deal with the TERM 'V' defined by '{y | y=y}' when the theory does not provide, for example, ExAy(yex <-> y=y). In Bernays class theory (and its basic variants) it's easy to say what V is. In that context, V is the class of all sets. But in ZF, there is no object of which every set is a member; so, in that context, IF 'V' is added to the actual language as a 0-place operation symbol, then, if we wish not to leave loose ends dangling ("terms with no reference"), then we confront how to deal with the term 'V'. MoeBlee
From: Marc Alcobé García on 5 Feb 2010 16:59 On 5 feb, 16:43, MoeBlee <jazzm...(a)hotmail.com> wrote: > On Feb 5, 6:56 am, Marc Alcobé García <malc...(a)gmail.com> wrote: > > > You said: > > > > it might help to be clear where the relativization method is > > > being used not to prove consistency itself, but rather RELATIVE > > > consistency > > > and also gave an example of statements of consistency and relative > > consistency. > > > My question is: when is a consistency proof using relativization not a > > relative consistency proof? > > When the theorem proved is not a relative consistency assertion but > rather a plain conistency assertion. > > > Can these kinds of proof be distinguished? > > Yes, by what the theorem proved is. If the theorem is a plain > consistency statement then the proof is a plain consistency proof. If > the theorem is a relative consistency statement, then the proof is > relative consistency proof. > > MoeBlee Maybe this ought to be obvious but I'll take the risk of asking: what distinguishes a plain consistency statement from relative consistency one? If T is any first-order theory we might ask if T is consistent. Then, I guess Con(T) could be considered as a plain consistency statement (at the level of the metatheory). To prove it, one can make use of set theory and try to find a structure where the (nonlogical) axioms of T are true. Now, if we try to do the same with a statement such as Con(ZF) we find ourselves using the very same theory whose consistency we are trying to prove. It is at this point that I really get lost.
From: MoeBlee on 5 Feb 2010 17:22
> Maybe this ought to be obvious but I'll take the risk of asking: what > distinguishes a plain consistency statement from relative consistency > one? I answered this twice already. A plain consistency statement is of the form: T is consistent. A relatvie consistncy statement is of the form: If T is consistent then T* is consistent. > If T is any first-order theory we might ask if T is consistent. Then, > I guess Con(T) could be considered as a plain consistency statement Yes, and a relative consistency statement is of the form: Con(T) -> Con(T*) > Now, if we try to do the same with a statement such as Con(ZF) we find > ourselves using the very same theory whose consistency we are trying > to prove. It is at this point that I really get lost. But we DON'T try to prove 'Con(ZF)' in ZF. MoeBlee |