The End of an Era - That of The Natural Numbers
in [Math]
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From: Nam Nguyen on 20 May 2010 23:33 William Hughes wrote: > On May 20, 1:19 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > > 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. > > Are you claiming that there is a model for an inconsistent > formal system? I already posted a subsequent post to clarify what I had said above. This is in that subsequent post (with a typo correction: "for all inconsistent theories", instead of "for all consistent theories"): >> There's a (minor) degree of glossing here. Technically, per >> each language L, there's one false model for all inconsistent >> theories written in that language. In details the false model >> per a language L(s1, S2, s3, ...) is: >> >> M = {<'A',U>, <=,{}>, <s1,{}>, <s2,{}>, <s3,{}>, ...} >> >> where U = {}, s1, s2, s3 are n-ary symbol of L. >> >> Having had the above caveat, there's only one kind of false models >> for all inconsistent theories: the kind in which all the U's and >> n-ary predicates are the empty set. So yes, an inconsistent formal system written in an L has the _false_ model per that L.
From: William Hughes on 20 May 2010 23:50 On May 21, 12:33 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > So yes, an inconsistent formal system written in an L has the _false_ > model per that L. Oh I see, an inconsistent system T does not have a model but it does have a false model. - William Hughes
From: Nam Nguyen on 21 May 2010 00:49 William Hughes wrote: > On May 21, 12:33 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> So yes, an inconsistent formal system written in an L has the _false_ >> model per that L. > > Oh I see, an inconsistent system T does not have a model but it > does have a false model. Right. But when we say about a consistent T's having a model we usually let it be understood that it's meant to be a model in which there's a non-empty relation for some formula to be true, unlike the false model. But either of the 2 model-types is still _a valid structure_ which is a _non-empty set of order-pairs_, each of which the 2nd component _might be an empty set_. Naturally. Don't fall into a trap of glossing over the phrase "a model" as if it meant there's only one model-type. That would be incorrect.
From: Nam Nguyen on 21 May 2010 01:12 Aatu Koskensilta wrote: > 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! I did a few times. Let F <-> (0 = x \/ 0 < x). Now on page 22, Shoenfield had something to the effect that the naturals is collectively a model of Q (he named it 'N'). Why don't you make some reflections of your own and tell us if F is true or false in the naturals, since you seem to believe the knowledge of the naturals is not of intuitive nature.
From: William Hughes on 21 May 2010 01:31
On May 21, 1:49 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > William Hughes wrote: > > On May 21, 12:33 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > >> So yes, an inconsistent formal system written in an L has the _false_ > >> model per that L. > > > Oh I see, an inconsistent system T does not have a model but it > > does have a false model. > > Right. So can you answer yes or no: Does an inconsistent system have a model? - William Hughes |