The End of an Era - That of The Natural Numbers
in [Math]
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From: Aatu Koskensilta on 20 May 2010 18:01 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > There's a model in which the universe and all n-ary relations > are empty, and this is the model for all inconsistent formal > systems. No it's not. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 20 May 2010 18:05 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > What's the point for me taking a course when I cited > _text book_ definition of model (e.g. condition iii pg 18, > phrase "other than =", Shoendfield, and other quotes), and > nobody _including you_ gave a slight reflection on them? You should reflect on Shoenfield's fine text more vigorously. Go on, reflect away! -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 20 May 2010 18:16 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Or if you'd would prefer the intuitions of Genzen over Hilbert's on > rules of inferences that's fine too. What intuitions of Gentzen and Hilbert would these be? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Alan Smaill on 20 May 2010 18:46 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Alan Smaill wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> Alan Smaill wrote: >>>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>>> >>>>> Alan Smaill wrote: >> ... >>>>>> It looks like you are claiming Shoenfield's formulation >>>>>> and yours are *equivalent* (after replacing "valid" with "true"). >>>>> On pg. 18 (on a "Structure", say, M) he had: >>>>> >>>>> "We want to define a formula A to be valid in M if all the meanings >>>>> of A are true in M". >>>>> >>>>> So he defined being "valid" as being "true". >>>> let's leave the terminology aside, and look at the logic. >>>> >>>>>> They are not: Shoenfield's version allows a formula to be valid even >>>>>> for an inconsistent T, and yours does not. >>>>> Where did he assert or stipulate that? >>>> When he said: >>>> >>>> "A formula is valid in T if it is valid in every model of T" >>> How does that invalidate a formula is being false in an inconsistent >>> T? >> >> By normal FOL reasoning; >> for example FOL with equality shows >> >> all x. (( x = 0 & x =/= 0 ) -> P(x)) >> >> for *any* predicate P. > > But where is the word "true" in all that syntactical, > rules-of-inference-based, proof? Nowhere. Let's try this one step at a time. Do you agree that all x. (( x = 0 & x =/= 0 ) -> P(x)) is provable in FOL with equality, whenever P is a unary predicate in the language under consideration? -- Alan Smaill
From: Nam Nguyen on 20 May 2010 23:13
Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Alan Smaill wrote: >>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>> >>>> Alan Smaill wrote: >>>>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>>>> >>>>>> Alan Smaill wrote: >>> ... >>>>>>> It looks like you are claiming Shoenfield's formulation >>>>>>> and yours are *equivalent* (after replacing "valid" with "true"). >>>>>> On pg. 18 (on a "Structure", say, M) he had: >>>>>> >>>>>> "We want to define a formula A to be valid in M if all the meanings >>>>>> of A are true in M". >>>>>> >>>>>> So he defined being "valid" as being "true". >>>>> let's leave the terminology aside, and look at the logic. >>>>> >>>>>>> They are not: Shoenfield's version allows a formula to be valid even >>>>>>> for an inconsistent T, and yours does not. >>>>>> Where did he assert or stipulate that? >>>>> When he said: >>>>> >>>>> "A formula is valid in T if it is valid in every model of T" >>>> How does that invalidate a formula is being false in an inconsistent >>>> T? >>> By normal FOL reasoning; >>> for example FOL with equality shows >>> >>> all x. (( x = 0 & x =/= 0 ) -> P(x)) >>> >>> for *any* predicate P. >> But where is the word "true" in all that syntactical, >> rules-of-inference-based, proof? > > Nowhere. > > Let's try this one step at a time. > > Do you agree that > > all x. (( x = 0 & x =/= 0 ) -> P(x)) > > is provable in FOL with equality, whenever P is a unary predicate > in the language under consideration? If by that you mean the meta statement {( x = 0 & x =/= 0 )} |- P(x) is true, then yes I agree. ("Provable" means provable in some formal system). |