The End of an Era - That of The Natural Numbers
in [Math]
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From: Marshall on 25 May 2010 20:08 On May 25, 11:29 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Aatu Koskensilta wrote: > > Nam Nguyen <namducngu...(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. They are the only thing that you understand. Marshall
From: Nam Nguyen on 25 May 2010 23:42 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: >>>>> >>>>>> 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,...}/ 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). > > (If you want to write <c1,0>, fine). > > 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? 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)? > > 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? >
From: Nam Nguyen on 25 May 2010 23:52 Jesse F. Hughes wrote: > 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? Why not, if this is the _degenerated structure_ (the false structure) of a language? Are you surprised that a tautology and a contradiction are equivalent in a _degenerated formal system_ that's called inconsistent? Surely you're not incapable of understanding that, are you?
From: Nam Nguyen on 25 May 2010 23:53 Marshall wrote: > On May 25, 11:29 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Aatu Koskensilta wrote: >>> Nam Nguyen <namducngu...(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. > > They are the only thing that you understand. It's more like it's the only thing _you_ could do in reasoning.
From: Nam Nguyen on 26 May 2010 00:06
Nam Nguyen wrote: > Alan Smaill wrote: >> >> 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). > >> >> (If you want to write <c1,0>, fine). >> >> 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? > > 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)? Note that FOL does provide us with non-formalized sets as a priori, and from which you could have finite numbers of set for your universe U, such as {}, {{}}, {{{}}}, if not more. |