The End of an Era - That of The Natural Numbers
in [Math]
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From: Nam Nguyen on 24 May 2010 20:22 Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> 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'. > > So far so strange. Strange in what way? Why don't you offer 2 examples of model: 1 for T1 and one of T2, then compare that with my M1, M2 and then point out the any "strangeness" that you saw? (Note you should use some notations for yours models). > >> 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. > > ???? Why don't we resolve the "so strange" issue you seem to have had above, before going on further. > >> 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_ ! > > Well, no doubt. > But when you say that you have a "model" for an inconsistent theory T, > then there are some conditions relating the axioms of T to the > structure defined so that M is in fact a model (that eaxh axiom > of T is true in M). Do you agree? > >> Note also the fact >> that per each language: there's only one inconsistent theory (as a collection >> of theorems) > > that's a normal outcome of eg Shoenfield's approach. > >> 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. > > Do you think gthis agrees with Shoenfield's definitions? > >> 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). > > You should not map a constant or function symbol to a > "predicate" or relation. Read what Shoenfield says about > the interpretation of constants (function symbols); > closed terms in the language should denote elements of the domain. > >>>> 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, U is *not* 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. >>> 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. > > Definition of truth of a formula involving a constant "e" requires > a non-empty domain to interpret e. 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. > > Well, no. > > If ever a discussion leads to the conclusion that there are no yes/no > questions, this one is it ..... > >
From: Nam Nguyen on 24 May 2010 20:28 William Hughes wrote: > On May 23, 3:38 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > <snip> > >> "(not(exits x) or blue(x))" isn't a well formed >> formula in L(T3), > > Ok if you do not want to use existential qualifiers > as predicates use the equivalent form > > For All x :(blue(x)) You're mistaken here. I have no preference on one quantifier over the other. It's just "(not(exits x) or blue(x))" isn't a wff, as I noted above and you still have NOT acknowledged! > > The important point is > >> How could any formula >> be true in a model in which its universe U and any >> n-ary predicate are empty? > > Note that by FOL any statement about the property > of the elements of an empty set is true. So use > the quantifier For All. Eg. > > For All x:(blue(x)) But _which definition_ of being true/false are you using here? (Note: FOL has more than one way to map a formula to a set of binary values!)
From: William Hughes on 24 May 2010 20:50 On May 24, 9:28 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > William Hughes wrote: > > On May 23, 3:38 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > > > <snip> > > >> "(not(exits x) or blue(x))" isn't a well formed > >> formula in L(T3), > > > Ok if you do not want to use existential qualifiers > > as predicates use the equivalent form > > > For All x :(blue(x)) > > You're mistaken here. I have no preference on one quantifier > over the other. It's just "(not(exits x) or blue(x))" isn't > a wff, as I noted above and you still have NOT acknowledged! > Mea culpa Mea culpa, Mea maxima culpa > > > > The important point is > > >> How could any formula > >> be true in a model in which its universe U and any > >> n-ary predicate are empty? > > > Note that by FOL any statement about the property > > of the elements of an empty set is true. So use > > the quantifier For All. Eg. > > > For All x:(blue(x)) > > But _which definition_ of being true/false are you using here? Yes it all depends what you mean by "true" I have decided that I will not use the meaning that other people use. From now on a sentence is true iff I say it is true. The question is who is to be master, that is all. > (Note: FOL has more than one way to map a formula to a set > of binary values!) This statement is false. - William Hughes
From: Nam Nguyen on 24 May 2010 21:05 William Hughes wrote: > On May 24, 9:28 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> But _which definition_ of being true/false are you using here? > > Yes it all depends what you mean by "true" > I have decided that I will not use the meaning that > other people use. From now on a sentence is > true iff I say it is true. The question is who > is to be master, that is all. I really don't know who the master be and I'm very much not interested in that. I'm just almost certain Tarski and Schoenfield (just to name two) didn't use the definition "a sentence is true iff I say it is true". >> (Note: FOL has more than one way to map a formula to a set >> of binary values!) > > This statement is false. In the theory T = {(x=x) /\ ~(x=x)), (x=x) and ~(x=x) are 2 equivalent formulas, right? Are they both true of both false? Or is one formula is true and an equivalent formula is false?
From: William Hughes on 24 May 2010 22:57
On May 24, 10:05 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > William Hughes wrote: > > On May 24, 9:28 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> But _which definition_ of being true/false are you using here? > > > Yes it all depends what you mean by "true" > > I have decided that I will not use the meaning that > > other people use. From now on a sentence is > > true iff I say it is true. The question is who > > is to be master, that is all. > > I really don't know who the master be When I say true it means exactly what I want it to mean, neither more nor less. > and I'm very much not > interested in that. I'm just almost certain Tarski and > Schoenfield (just to name two) didn't use the definition > "a sentence is true iff I say it is true". This statement is false. - William Hughes |