The End of an Era - That of The Natural Numbers
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From: Nam Nguyen on 21 May 2010 03:47 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> I'm sure you've heard of "equivalent formulas" in a theory. > > What of them? > Aren't 2 equivalent formulas either both true or both false?
From: William Hughes on 21 May 2010 07:09 On May 21, 4:43 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Is the formula Exy[~(x=y)] provable? That's a yes or no > question, in your style of questioning, right? So answer > Yes, There is a theory T1 where it is provable, so it is provable. The fact that there is a theory T2 where it is not provable does not change this. Your turn Let T be an inconsistent system. You note that a trivial model exists, call it L (the existence of L is not disputed). You have stated that L is a "false model" for T (despite the fact that there are formula provable in T that are true in L, and formula provable in T that are false in L). We know what L is but we do not know what a "false model" is. In particuar, we do not know if the fact that "T has a 'false model'" implies that T has a model. The simple "yes/no" question is Is there a model L1 with the property that every provable formula in T is true in L1? - William
From: Nam Nguyen on 21 May 2010 23:15 William Hughes wrote: > On May 21, 4:43 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> Is the formula Exy[~(x=y)] provable? That's a yes or no >> question, in your style of questioning, right? So answer >> > > Yes, There is a theory T1 where it is provable, so it is > provable. The fact that there is a theory T2 where > it is not provable does not change this. Oh. So it isn't just a simple "yes/no" question anymore! _You had to explain it beside a yes/no_! > > Your turn Sure. My turn to answer a non-"yes/no" question. > > Let T be an inconsistent system. You note that a trivial > model exists, call it L (the existence of L is not > disputed). Ok. You're no more disputing with me a false model - nonetheless a model - exists for an inconsistent T. > You have stated that L > is a "false model" for T (despite the fact that there > are formula provable in T that are true in L, and formula > provable in T that are false in L). _You_ have yet to prove, according to the definition of model, a formula to be true in a model whose universe and all n-ary predicates are empty! Until you do so, you can't assert "there are formula provable in T that are true in L". > We know what L is > but we do not know what a "false model" is. Above you yourself said "a trivial model exists, call it L", and I have explained to you more than one time each language L(T) has that trivial model L whose universe and all n-ary predicates are empty. What did you mean that you "do not know what a 'false model' is"? > In particuar, > we do not know if the fact that "T has a 'false model'" > implies that T has a model. So, why did you say L as "a trivial model exists, call it L"? > > The simple "yes/no" question is > > Is there a model L1 with the property that > every provable formula in T is true in L1? That's not what you asked me before and, iirc, I didn't say of such a thing. But my answer to this question would be no, having not reflected too much on it yet.
From: William Hughes on 21 May 2010 23:57 On May 22, 12:15 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > William Hughes wrote: > > The simple "yes/no" question is > > > Is there a model L1 with the property that > > every provable formula in T is true in L1? > > That's not what you asked me before and, iirc, I didn't say of > such a thing. > > But my answer to this question would be no, having not reflected > too much on it yet. In the sense that the rest of the world means "model" an inconsistent T does not have a model. Next question. Let L be the trivial model. Is "for all x, x=/=x" true in L. - William Hughes
From: Nam Nguyen on 22 May 2010 00:38
Nam Nguyen wrote: > William Hughes wrote: Let me give you an example step by step so you could understand the concept of the false model of an inconsistent theory. Let "blue" be an unary predicate symbol of a language that also has an individual constant "e", and let: T1 = {P(e)} be consistent with this model M1: M1 = { <'A',U>, <'e',(e0}>, <'=',{<e0,e0>}>, <'~=',{}>, <'blue',{e0}>, <'~blue',{}> } where e0 = {}, U = {e0}, '~=' and '~blue' are a (FOL) defined symbols for the unary predicates != and ~blue(x), respectively. Let now: T2 = {~P(e)} M2 = { <'A',U>, <'e',(e0}>, <'=',{<e0,e0>}>, <'~=',{}>, <'blue',{}>, <'~blue',{e0}> } Now let T3 be our intended theory T3 = {P(e) /\ ~P(e)} The false model for T3 is: M3 = { <'A',{}>, <'e',(}>, <'=',{}>, <'~=',{}>, <'blue',{}>, <'~blue',{}> } where now U = {}. Note that both the negating predicates 'blue' and '~blue', as well as the predicate '=' are empty sets, but despite that M3 can never be empty! So, by definition of being true, being false in FOL model, all formulas are are defined as being false in M3, and that's the only model for T3 - the false model. |