The End of an Era - That of The Natural Numbers
in [Math]
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From: Aatu Koskensilta on 25 May 2010 12:57 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Other than that, I'm afraid any conversation I might have with you > would be fruitless. Well, yes, as already noted I've at long last concluded it's totally pointless to try to discuss logic with you, my contributions thus reduced to general observations and cheap pot-shots. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 25 May 2010 14:27 Alan Smaill wrote: > 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.) You'd still believe it's strange and not following Shoenfield's convention. OK. Why don't you read pg. 9-10 on his definitions of "n-tuple", "n-ary predicate", and explain why for 2-tuple the notation of ordered pair <> I've used is "strange"? 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. > > 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. Why don't you construct such an M4 for T4 and then I'll explain again/further my notations as well as my claims about some formulas being true/false or what not. (Other than that, I'm afraid here I couldn't explain things with just English wordings alone, and our conversation wouldn't be able to move forward).
From: Nam Nguyen on 25 May 2010 14:29 Aatu Koskensilta wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Other than that, I'm afraid any conversation I might have with you >> would be fruitless. > > Well, yes, as already noted I've at long last concluded it's totally > pointless to try to discuss logic with you, my contributions thus > reduced to general observations and cheap pot-shots. > I guess cheap shots will be flying around then. What's really new?
From: Alan Smaill on 25 May 2010 14:41 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: >>>> >>>>> 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.) > > You'd still believe it's strange and not following Shoenfield's > convention. OK. Why don't you read pg. 9-10 on his definitions > of "n-tuple", "n-ary predicate", and explain why for 2-tuple > the notation of ordered pair <> I've used is "strange"? Nothing wrong wrong with that -- no complaint there. > 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,...}/ Interpretation of constants: c1 |-> 0, c2 |-> 1. (If you want to write <c1,0>, fine). Predicate blue map to relation R, where R(x) <-> x = 1 So, no separate definition for "~ blue" since it is defined uniformaly for all interpretations following Tarski. No separate definition for the universal quantifier, either. >> 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. > > Why don't you construct such an M4 for T4 and then I'll explain > again/further my notations as well as my claims about some formulas > being true/false or what not. (Other than that, I'm afraid here I > couldn't explain things with just English wordings alone, and our > conversation wouldn't be able to move forward). Done. 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: Jesse F. Hughes on 25 May 2010 14:50
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Aatu Koskensilta wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> Other than that, I'm afraid any conversation I might have with you >>> would be fruitless. >> >> Well, yes, as already noted I've at long last concluded it's totally >> pointless to try to discuss logic with you, my contributions thus >> reduced to general observations and cheap pot-shots. >> > > I guess cheap shots will be flying around then. What else is possible, when the topic is logic, but one of the conversants is a blowhard incapable of realizing that, whenever P is false in a structure, ~P is true in that same structure? -- Jesse F. Hughes "What do you tremble your *soul* before it for?" he cried. "You don't learn algebra with your blessed soul. Can't you look at it with your clear simple wits?" -- D.H. Lawrence, /Sons And Lovers/ |