The End of an Era - That of The Natural Numbers
in [Math]
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From: Nam Nguyen on 24 May 2010 23:19 William Hughes wrote: > 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
From: Nam Nguyen on 25 May 2010 00:26 Jesse F. Hughes wrote: > 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. <Caveat> This isn't a conversation of mine to Jesse F. Hughes but is just a comment on some technical point mentioned in a public post. </Caveat> Jesse wasn't correct on what Shoenfield wrote on pg. 19, where Schoenfield wrote: "If A is ~B, then M(A) is H_~(M(B)) where (on the same pg. 19) M(A) = T iff pM(M(a1), ..., M(an)) and "(i.e., iff the n-tuple (M(a1), ..., M(an)) belongs to the predicate pM". [where M here is a structure]. In other words, on pg. 19, Jesse seems to have forgot or ignored the fact that Schoenfield was using the sense of being true as the sense of _being true in a model_. [After all, pg. 19 is a key part of his Section 2.5 "Structure"]. As well, on pg. 12 _only_ general truth functions/mentioned are defined and this isn't a section about models. > > He reflected the bejeezus out of those pages, until in despair, they > said just what he wanted them to. > >
From: Alan Smaill on 25 May 2010 05:01 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > 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? They are both valid in T, according to Shoenfield's definition. -- Alan Smaill
From: Alan Smaill on 25 May 2010 05:36 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: >>>> >>>>> 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). Strange in that it is not the normal way of going about it. (It doesn't follow it's wrong, but if you think you are following Shoenfield, then it would be easier to follow for others, and maybe yourself, to keep to a standard presentation, rather than invent your own.) A more conventional notation supplies: (i) the domain of the structure (a set, U in your case), (ii) denotations for constants (elements of U) (iii) same for function symbols (iv) and n-ary relation for each n-ary predicate. Then use Tarski's definition. >>> 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. The ???? refers to your statement: "<'s2',p2> and <'s2',p2> have been used interchangeably". The questions below remain: >>> 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 this agrees with Shoenfield's definitions? Do you think this agrees with Shoenfield's definitions? >>> 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. Do you accept this? >> Read what Shoenfield says about >> the interpretation of constants (function symbols); >> closed terms in the language should denote elements of the domain. >> Definition of truth of a formula involving a constant "e" requires >> a non-empty domain to interpret e. That's a text-book definition. Do you accept this? -- Alan Smaill
From: Jesse F. Hughes on 25 May 2010 07:35
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Jesse F. Hughes wrote: >> 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. > > <Caveat> > This isn't a conversation of mine to Jesse F. Hughes but is just > a comment on some technical point mentioned in a public post. > </Caveat> > > Jesse wasn't correct on what Shoenfield wrote on pg. 19, where > Schoenfield wrote: > > "If A is ~B, then M(A) is H_~(M(B)) > > where (on the same pg. 19) M(A) = T iff pM(M(a1), ..., M(an)) > and "(i.e., iff the n-tuple (M(a1), ..., M(an)) belongs to the > predicate pM". [where M here is a structure]. You're absolutely right. I said he defined the truth value of ~ B as H_~(B), when it should be H_~(M(B)). (Actually, what Nam writes as M is a funky script A in my copy, but no matter.) > In other words, on pg. 19, Jesse seems to have forgot or ignored > the fact that Schoenfield was using the sense of being true as > the sense of _being true in a model_. [After all, pg. 19 is a key > part of his Section 2.5 "Structure"]. Er, no, I forgot nothing of the sort. You're *also* talking about being true in a structure, namely the structure with empty support. > As well, on pg. 12 _only_ general truth functions/mentioned are > defined and this isn't a section about models. So? Page 12 is where he defines H_~. That definition applies on p. 19, too. Look, evidently you're just too stupid to get my point, but here it is. You say that in the empty structure, every formula evaluates to F. This is, of course, silly, since whenever a formula B evaluates to F in a structure, the formula ~B evaluates to T *by definition* of truth value in a structure. It's right there on p. 19, as clear as day. I'll say it again slowly. In *any* structure M, and for *any* formula B, if B evaluates to F in M, then ~B evaluates to T. Thus, it cannot be the case that every formula is F in some structure. (Here I ignore the fact that Shoenfield explicitly defines a structure to have non-empty support and thus Nam's empty "model" is not a structure at all.) -- "I deal with reality. It's a brutal reality. But it's the only one we've got. And people like me, do what it takes. I'm part of a long line of discoverers. So I do what it takes." -- James S. Harris channels George W. Bush |