The End of an Era - That of The Natural Numbers
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From: Aatu Koskensilta on 24 May 2010 10:40 stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: > Then to me it makes sense to say that D' is a model of Th(D'), > even in the case where D' is the empty set. Yes, this is an example of a general result that becomes more awkward to state on the requirement that models have non-empty domains. -- 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 24 May 2010 11:26 Marshall <marshall.spight(a)gmail.com> writes: > I recall reading somewhere that a model has to have at least two > elements in the domain. I thought it weird, so it stuck in my head. Never seen this -- unless it's something from Boolean Algebra where some (most?) but not all authors insist one way or another that 0=/=1. > > > Marshall -- Alan Smaill
From: Nam Nguyen on 24 May 2010 14:22 Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> 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',{}> >> } > > This is a strange way to give a model -- e.g. the interpretation of the > negation of the predicate "blue" is determined by the general > definition of satisfaction, so it is confusing to specify it here. > >> where e0 = {}, U = {e0}, '~=' and '~blue' are a (FOL) defined symbols >> for the unary predicates != and ~blue(x), respectively. > > What is A? 'A' is, irrc, a notation from Herbert B. Enderton indicating the universal quantification over the universe U of discourse, another poster (MoeBlee) in a past thread mentioned and used the notation. Shoenfield didn't' use it iirc, but it won't matter much in the definition of model (a la Tarski). The general definition of structure/model M (of a language), which a model of a theory is, would be: M = {<>, <>, <>, <>, <>, <>, ....} or in some details noting we're in FOL=: M = {<'A',U>, <'=',p0>, <'s1',p1>, <'s2',p2>, <'s3', p3>, ...} Where 'A' is the Universal Quantification symbol, and each 'si' is an n-ary predicate symbol of the language L, and each pi is the predicate (a set) corresponding to si. Note that an individual constant symbol, such as 'e' of L(T3), is a 0-ary symbol. Also, e0 is just 0-ary function value for 'e', meaning e0 is an element of U that's named 'e'. Note also that for the 2nd element of an ordered pair <>, sometimes I close them in brackets not at other times: e.g. <'s2',p2> and <'s2',p2> have been used interchangeably. But that'd be just notation. The key feature of this definition I think my opponents have overlooked is ultimately M is a _set_ and as such any language will have to have a model: _M might be empty but M will exist as a set_ ! Note also the fact that per each language: there's only one inconsistent theory (as a collection of theorems) and there's is only one model that all the the n-ary predicates are empty. That one model (a set) is the model for that one theory. What my opponents seem to get confused is that the meta expression "There's no model for an inconsistent theory T" is actual wrong: there's always that peculiar model (which again _is a set_): it might be the only useless model; but that's why it's a model for the only useless theory in that language! >> 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',{}> >> } > > Stop right there. > You have > > <'e',(}> > > (I assume you mean <'e',{}>). > > But the interpretation of 'e' has to be a *member* of the domain of > interpretation, in the usual version of model, as in Shoenfield. You > have broken that part of the definition. The textbook definition of model (Shoenfield's included) follows Tarski's concept of truth and makes use of the property "being in, being a member of" with respect to a predicate, an ordered pair of elements in the universe U. But since both the predicate p and U can be empty set, there's nothing with <'e',{}>: by definition of a structure, you have to _map_ a symbol ('e' in this case) to a predicate ({}an empty set in this case). > >> 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! > > And what entity does "e0" refer to? As mentioned above that's a named element of U that gets map to the language symbol 'e' > >> 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. > > Not by the usual definition of being true or false in FOL model. That's not correct. Definition of a formula being true or false in M requires ONLY the notion of the n-tuples of elements of U _factually_ be in or not in M. (And M always exists as a model-set!). And that's a text-book definition. If it would help, one doesn't have to like the word the "the false model", one just have to acknowledge that by definition of a structure (a set) there's such a model that using the membership property and following Tarski's concept of truth all formulas are interpreted as false.
From: Aatu Koskensilta on 24 May 2010 14:35 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > If it would help, one doesn't have to like the word the "the false > model", one just have to acknowledge that by definition of a structure > (a set) there's such a model that using the membership property and > following Tarski's concept of truth all formulas are interpreted as > false. Were you perhaps led to this important discovery by vigorously reflecting on Shoenfield? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Jesse F. Hughes on 24 May 2010 14:50
Aatu Koskensilta <aatu.koskensilta(a)uta.fi> writes: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> If it would help, one doesn't have to like the word the "the false >> model", one just have to acknowledge that by definition of a structure >> (a set) there's such a model that using the membership property and >> following Tarski's concept of truth all formulas are interpreted as >> false. > > Were you perhaps led to this important discovery by vigorously > reflecting on Shoenfield? He reflected especially hard on p. 19, where Shoenfield defined the truth value of ~ B as H_~(B), and on p. 12 where Shoenfield defined H_~(F) = T. He reflected the bejeezus out of those pages, until in despair, they said just what he wanted them to. -- Jesse F. Hughes "I wish I could have been around when the Founding Fathers penned the 9th amendment to help them out." --Archimedes Plutonium, political scientist, legal scholar |