The End of an Era - That of The Natural Numbers
in [Math]
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From: Marshall on 26 May 2010 08:24 On May 26, 5:18 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Marshall says... > > > > >On May 26, 4:09=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) > >wrote: > > >> So "There does not exist an x such that blue(x)" is true > >> for the empty model, while "There does not exist an x such > >> that blue(x)" is false. But both sentences are valid for > >> the empty model. > > >This paragraph seems to have a typo in it. > > Doh! The second sentence was supposed to be "There exists an > x such that blue(x)", the negation of the first sentence. Ah! That makes perfect sense. > A typo like that makes an entire post incomprehensible. It's not that bad. You had four perfectly clear and comprehensible paragraphs, and once obviously broken sentence which you promptly fixed. Marshall
From: Daryl McCullough on 26 May 2010 08:25 Daryl McCullough says... >So "There does not exist an x such that blue(x)" is true >for the empty model, while "There does not exist an x such >that blue(x)" is false. But both sentences are valid for >the empty model. That should have been: So "There does not exist an x such that blue(x)" is true for the empty model, while "There exists an x such that blue(x)" is false. But both sentences are valid for the empty model. -- Daryl McCullough Ithaca, NY
From: Alan Smaill on 26 May 2010 11:11 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Alan Smaill wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> .... >>> In fact, why don't you yourself _literally spell-out (present)_ >>> a model M4 for the below very simple T4, using only technical >>> notations, for n-tuples or what not. (M4 doesn't have to be infinite). >>> >>> T4 = {~blue(c1}}, where L(T4) = L(c1,c2,blue), where 'c1', 'c2' >>> are individual constants and 'blue' is an 1-ary predicate symbol. >> >> Domain: natural numbers, {0,1,2,...}/ > > So that's just a the universe U in Shoenfield's terminology. > (I think usually the term "domain" is reserved for mapping or > function; and U is just a set). > >> >> Interpretation of constants: c1 |-> 0, >> c2 |-> 1. > > And that's just <'c1',{0}> and <'c2',{1}> which isn't that > different from what I had ( and which you alluded to right > below). No, c1 does not denote a set " {0} ", it denotes the number 0, as I said: >> (If you want to write <c1,0>, fine). Remember, c1 denotes an *element* of Universe, not a subset of the universe. >> Predicate blue map to relation R, where >> >> R(x) <-> x = 1 > > My mistake here was that instead only mildly suggesting > "(M4 doesn't have to be infinite)", I should have outright > asked for a finite example that also does NOT depend on another > model, such as the naturals as the standard model of arithmetic. > I mean after all your example should clarify Shoenfield's > definition of model but it has a degree of circularity: did > you spell out the model known as the natural numbers? what > exactly "..." mean in your universe U? Fine, take U = {0,1,2}, and take everything else as above. > In any rate, _there's a finite model_ for T4 and I was requesting > for a "literally spell-out" of the model, in the sense of literally > listing the model out. Could you perhaps present a finite model > by listing out all necessary mappings between language symbols > and predicates (sets)? Are you claiming that your notion of model is equivalent to Shoenfield's? Do you agree that it follows from his definition that a constant is interpreted as an element of the domain, and that therefore the domain is not empty whenever there is a constant in the language? -- Alan Smaill
From: Nam Nguyen on 27 May 2010 02:05 Alan Smaill wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Alan Smaill wrote: >>> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>> > ... >>>> In fact, why don't you yourself _literally spell-out (present)_ >>>> a model M4 for the below very simple T4, using only technical >>>> notations, for n-tuples or what not. (M4 doesn't have to be infinite). >>>> >>>> T4 = {~blue(c1}}, where L(T4) = L(c1,c2,blue), where 'c1', 'c2' >>>> are individual constants and 'blue' is an 1-ary predicate symbol. >>> Domain: natural numbers, {0,1,2,...}/ >> So that's just a the universe U in Shoenfield's terminology. >> (I think usually the term "domain" is reserved for mapping or >> function; and U is just a set). >> >>> Interpretation of constants: c1 |-> 0, >>> c2 |-> 1. >> And that's just <'c1',{0}> and <'c2',{1}> which isn't that >> different from what I had ( and which you alluded to right >> below). > > No, c1 does not denote a set " {0} ", it denotes the number 0, > as I said: > >>> (If you want to write <c1,0>, fine). > > Remember, c1 denotes an *element* of Universe, not a subset of > the universe. It's all mapping (association) and I was thinking about 1-1 mapping between 0 and {0}, but sure I'll stay with the convention and use 0 instead of (0}. > >>> Predicate blue map to relation R, where >>> >>> R(x) <-> x = 1 >> My mistake here was that instead only mildly suggesting >> "(M4 doesn't have to be infinite)", I should have outright >> asked for a finite example that also does NOT depend on another >> model, such as the naturals as the standard model of arithmetic. >> I mean after all your example should clarify Shoenfield's >> definition of model but it has a degree of circularity: did >> you spell out the model known as the natural numbers? what >> exactly "..." mean in your universe U? > > Fine, take U = {0,1,2}, and take everything else as above. Well, so far you've only spelled out U (and in effect <'A',U>). You've not spelled out the mapping (ordered pair) <'blue',p_blue> where p_blue is an actual _set_. Iow, if R is p_blue, can you spell out the predicate-set R? >> In any rate, _there's a finite model_ for T4 and I was requesting >> for a "literally spell-out" of the model, in the sense of literally >> listing the model out. Could you perhaps present a finite model >> by listing out all necessary mappings between language symbols >> and predicates (sets)? > > Are you claiming that your notion of model is equivalent to > Shoenfield's? That's a question/request put forward to you: not a claim. (I mean, his definition should support a spelling out of a very small finite model, right?) > > Do you agree that it follows from his definition that a constant > is interpreted as an element of the domain, Suppose you have a theory T5 = {Ax[~(x=e)]}, which element of your "domain" U (whatever U might be) would get interpreted as e? > and that therefore > the domain is not empty whenever there is a constant in the > language? In his definition, as I had before, his structure is a non-empty set of ordered pairs, in each of which the 2nd component _is a set_ (un-formalized kind of set that should be taken a priori but which nonetheless could be empty).
From: Alan Smaill on 27 May 2010 07:34
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: >>>> >> ... >>>>> In fact, why don't you yourself _literally spell-out (present)_ >>>>> a model M4 for the below very simple T4, using only technical >>>>> notations, for n-tuples or what not. (M4 doesn't have to be infinite). >>>>> >>>>> T4 = {~blue(c1}}, where L(T4) = L(c1,c2,blue), where 'c1', 'c2' >>>>> are individual constants and 'blue' is an 1-ary predicate symbol. >>>> Domain: natural numbers, {0,1,2,...}/ >>> So that's just a the universe U in Shoenfield's terminology. >>> (I think usually the term "domain" is reserved for mapping or >>> function; and U is just a set). >>> >>>> Interpretation of constants: c1 |-> 0, >>>> c2 |-> 1. >>> And that's just <'c1',{0}> and <'c2',{1}> which isn't that >>> different from what I had ( and which you alluded to right >>> below). >> >> No, c1 does not denote a set " {0} ", it denotes the number 0, >> as I said: >> >>>> (If you want to write <c1,0>, fine). >> >> Remember, c1 denotes an *element* of Universe, not a subset of the >> universe. > > It's all mapping (association) and I was thinking about 1-1 mapping > between 0 and {0}, but sure I'll stay with the convention and use > 0 instead of (0}. OK >>>> Predicate blue map to relation R, where >>>> >>>> R(x) <-> x = 1 >>> My mistake here was that instead only mildly suggesting >>> "(M4 doesn't have to be infinite)", I should have outright >>> asked for a finite example that also does NOT depend on another >>> model, such as the naturals as the standard model of arithmetic. >>> I mean after all your example should clarify Shoenfield's >>> definition of model but it has a degree of circularity: did >>> you spell out the model known as the natural numbers? what >>> exactly "..." mean in your universe U? >> >> Fine, take U = {0,1,2}, and take everything else as above. > > Well, so far you've only spelled out U (and in effect <'A',U>). > You've not spelled out the mapping (ordered pair) <'blue',p_blue> > where p_blue is an actual _set_. Iow, if R is p_blue, can you > spell out the predicate-set R? The set is the extension of the relation R: { x in U | R(x) } = {1} >>> In any rate, _there's a finite model_ for T4 and I was requesting >>> for a "literally spell-out" of the model, in the sense of literally >>> listing the model out. Could you perhaps present a finite model >>> by listing out all necessary mappings between language symbols >>> and predicates (sets)? >> >> Are you claiming that your notion of model is equivalent to >> Shoenfield's? > > That's a question/request put forward to you: not a claim. > (I mean, his definition should support a spelling out of a > very small finite model, right?) Of course it does. But it's a question to you also, when you claim that you have a model of an inconsistent theory. Are you claiming that your notion of model is equivalent to Shoenfield's? >> Do you agree that it follows from his definition that a constant >> is interpreted as an element of the domain, > > Suppose you have a theory T5 = {Ax[~(x=e)]}, which element of your > "domain" U (whatever U might be) would get interpreted as e? Could be any object in U. But whatever it is, the structure is not a model for T5, i.e. it will not satisfy the statement Ax[~(x=e)]. Just follow Tarski's definition. >> and that therefore >> the domain is not empty whenever there is a constant in the >> language? > > In his definition, as I had before, his structure is a non-empty > set of ordered pairs, in each of which the 2nd component _is a set_ > (un-formalized kind of set that should be taken a priori but which > nonetheless could be empty). That's for predicates, not for constants. Look at his treatment of constants (& function symbols). -- Alan Smaill |