Models of set theory vs. models of first order theories
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From: Marc Alcobé García on 6 Feb 2010 03:08 On 5 feb, 23:22, MoeBlee <jazzm...(a)hotmail.com> wrote: > > 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 But to prove Con(T) you already need the consistency of something else... don't you?
From: Marc Alcobé García on 6 Feb 2010 03:23 On 5 feb, 23:22, MoeBlee <jazzm...(a)hotmail.com> wrote: > > 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 Already, Kunen tries to give a platonistically sensible "proof" of Con(ZF). In the chapter on relative consistency proofs where he talks about reflection theorems. I wonder how is that possible... Because I think that Con(ZF) requires Con(Somethin else) to be proved like Con(ZF-Foundation), for example. I don't know to make that distinction between plain and relative. Maybe I am confusing different levels (theory vs. metatheory for example).
From: Nam Nguyen on 6 Feb 2010 03:49 Marc Alcob� Garc�a wrote: > On 5 feb, 23:22, MoeBlee <jazzm...(a)hotmail.com> wrote: >>> 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 > > But to prove Con(T) you already need the consistency of something > else... don't you? If by "prove" we meant "syntactically prove by inference rules" then an inconsistency would also prove Cont(T).
From: Nam Nguyen on 6 Feb 2010 03:53 Nam Nguyen wrote: > Marc Alcob� Garc�a wrote: >> On 5 feb, 23:22, MoeBlee <jazzm...(a)hotmail.com> wrote: >>>> 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 >> >> But to prove Con(T) you already need the consistency of something >> else... don't you? > > If by "prove" we meant "syntactically prove by inference rules" then > an inconsistency would also prove Cont(T). All of this means from the syntactical, rule-of-inference, point of view, there's just no way to prove any Con(T), unless an inconsistency is involved.
From: Marc Alcobé García on 8 Feb 2010 04:08
> But to prove Con(T) you already need the consistency of something > else... don't you? Clearly there are two uses of the word "model". One is to refer to structures for first-order languages which make true a set of formulas in a given language (the nonlogical axioms of a theory). Structures are anything but sets. The other one is to refer to relative interpretations of a theory T in another theory T'. In this last case all we can say is that if T' has a model, then T has a model, thereby obtaining only a relative consistency proof. Now, the question can be rewritten as: is any statement of the form Con(T) a statement in a given language of a certain theory T' (the metatheory) which interprets T (the theory)? Then, our proof of Con(T) would be only a proof of Con(T') -> Con(T). I think that this might be one of the lessons of Gödel's incompleteness theorems, i. e. that since elementary arithmetic is powerful enough to interpret its own metatheory, there is no way we can prove its consistency (unless we use a stronger metatheory that elementary arithmetic cannot interpret). Does any of this make any sense? |