The End of an Era - That of The Natural Numbers
in [Math]
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From: Nam Nguyen on 17 Jun 2010 23:54 MoeBlee wrote: > On Jun 16, 8:27 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> MoeBlee wrote: >>> On Jun 15, 11:13 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> Yes it's a fact that he chose to narrow his definition to restrict >> U to be non-empty. > > I wouldn't say he narrowed, since it is the usual definition. But the usual definition of set doesn't exclude a set that's empty. So, if we choose set as a means for validating, say, "Material Adequacy" of some concept (of being true of a formula in this case) then some narrowing of set definition has occurred. > But, in > the sense that his definition is narrower than some other definitions > (such as in free logic, as I understand), okay. > >>> If one wishes to >>> consider a notion of a structure for a language in which the universe >>> for the stucture is empty, then one must revise Shoenfield's >>> definition and recognize that one is then working with a different >>> definition from Shoenfield's. >> I don't agree. > > If you say the universe can be non-empty, then you're working with a > different definition as Shoenfield. As I see it, I was examining the case where U is empty and Shoenfield wasn't examining that case; and each had a different purpose in mind: he was writing a textbook and I was reviewing issues in the foundation. As I mentioned in the isosceles triangle analogy, if he had to narrow a definition to fit better his case then so be it. But both he and I still used Tarski's Semantic Concept Of Truth ("Tscot", for short). Note that naturally "Semantic Concept Of Truth" isn't something I coined. > And to work with your definition > we must recognize that it is revised from Shoenfields in at least the > sense that you allow empty universes. I actually don't have any new definition, viz-aviz Tcot. It's Shoenfield who had a restricted definition which I think is due to the scope of his writing a textbook. > > Surely, you agree with this? Since even Shoenfield and I were examing 2 opposite cases I still used other people's definition of truth (Tscot), I can't agree here. (Though I don't think my disagreement here matters much as far as the debate itself is concerned.) > >> Each author's textbook is apt to have some _peculiarities_ >> that other authors might or might not have. Shoenfield, e.g., defined >> a formula being "valid" in T instead of being "true" in T. > > Of course, fair enough. > > Except that Shoenfield's stipulation of non-empty universe isn't > peculiar. It's virtually part of any ordinary definition in > mathematical logic. Of course, there are defintions in which the > universe may be non-empty, and we may study such things, but in > ordinary mathematical logic, the universe is stipulated (i.e., part of > the defintion of 'structure for a language') to be non-empty. But what some people don't seem to have realized is that such peculiarity has cost them their understanding why x=x isn't true in _all contexts of FOL_ reasoning. Logical reasoning framework such as FOL is supposed to be "tight", virtually "perfect": any peculiarity is apt to mean subtlety, hidden pitfall, that one could fall into if not being careful. To be precise, whether or not x=x is true _in all cases_ definition of being true for a formula should be equally in existence _for all cases_. After all, "ordinary mathematical logic" or not we're still talking about the same FOL= and in a single context, if a formula is defined, e.g., as being true then such definition has to be _uniformly applicable in all cases_. (Though again one is free to narrow the scope of discussion to only 1 case. But in our debate, such isn't the case). >> >> Well, "_within the context of FOL reasoning_" means the subject we've been >> debating is at a level above any specific peculiarities of a given author's >> writing. >> >> So no, I don't agree Shoenfield's book needs to be rewritten, notwithstanding >> he had a restriction on the universe U. > > I didn't say his book needs to be rewritten. I'd just like to be clear > that you agree with the following: > > If you say the universe can be non-empty, then you're working with a > different definition from Shoenfield. Yes, I agree. His definition is applicable only where U is empty, so his definition is narrower (hence different) from the one I'm using (Tscot). > And to work with your definition > we must recognize that it is revised from Shoenfield's in at least the > sense that you allow empty universes. Again, I don't have any new definition: I'm just using an existing definition in a different case (where U is empty). And the definition I'm using requires an object language (to write formulas under question such as x=x), a metalanguage to define truth and to evaluate "Material Adequacy" for truth assignment, etc.... But such definition doesn't at all have any requirement that U _must_ be non-empty. But yes, his definition is different from what I've been using. In fact I already alluded to it before that his definition is subsumed in a bigger definition of a formula being model theoretically true/false a la Tscot. > > Saying 'yes' to that is not a big philosophical deal. Just common > sense that allows us to communicate. Right? Any thing that allow us to communicate is good with me as well.
From: Nam Nguyen on 18 Jun 2010 00:15 Jesse F. Hughes wrote: > MoeBlee <jazzmobe(a)hotmail.com> writes: > >> But if the universe is empty while the set of constants is non-empty >> then there is no such function. >> >> So, I don't see how we can have an empty universe for a structure for >> a language that has constants. > > Of course that's right. > > And that's how it *should* be. In reasoning framework, in the end we'll go only by _definition_ and permitted inferences. Feelings such "Of course that's right" or how thing "should/shouldn't" be would be at best just intuitions which more often than not turn out to be wrong. > If a theory has constants, then its > structures must be non-empty. No (model) structure within FOL= can be the empty _set_. It has at least these 2 elements: <'A',U> and <'=',{all 2-tuples of some form} | empty set> _by definition_.
From: Nam Nguyen on 18 Jun 2010 01:35 MoeBlee wrote: > On Jun 17, 5:02 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> Nam Nguyen <namducngu...(a)shaw.ca> writes: >>> So what's in you mind the definition of x=x being true be, model >>> theoretically speaking? Can you _explicitly_ state the definition? >> Just read some more Shoenfield. > > Wait, if 'x' is a variable, then, at least by pg. 19, there is no > truth value for "x=x". That's actually not true, in closer inspection. What might make one easily confused on the issue is he actually refering to 2 different formulas: the original L in question, and the L(fancyA). Then he said (pg. 19): "We shall now define a truth value fancyA(A) for each closed formula in L(fancyA)"! So the closed formula A is in L(fancyA). So x=x is not in L(fancyA), an extension of L, in the sense that x=x is already in L. > That is, Shoenfield, gives truth values only > for closed formulas (sentences), and he bypasses defining > 'satisfaction of a formula' (for formulas in general including open > formulas) with a certain technique. Right? No. He actually defined A (of L) being true on pg. 18, and pg. 19, where A is _either_ closed or open! [On pg. 19, this is in the paragraph before the last lemma.] > > Also, Aatu, I haven't been following this thread in every detail. That > said, if I'm not mistaken there is one glitch that has to be addressed > in allowing a version of Shoenfield with empty domains. > > In Shoenfield's definition of a structure, (when truly formalized) > there is a function from the set of constants into the universe: > > "...for each constant e of L, e_fancyA is an individual of fancyA." > > I take that formally to mean there is a function f from the set of > constants into the universe of fancyA (universe of fancy_A = | > fancyA|): > > f(e) = e_fancyA > and > e_fancyA in |fancyA| > > So there must exist f such that > > f:set_of-constants -> |fancyA| > > But if the universe is empty while the set of constants is non-empty > then there is no such function. > > So, I don't see how we can have an empty universe for a structure for > a language that has constants. In such case, you just map any language constant C to the empty set {}, to denote there's no (named) element in U to associate C with. > > Meanwhile, let me see whether I understand the heart of the discussion > correctly: > > The truth value of Ax x=x (where 'x' is a variable) is T iff > > the truth value of x=x _x[i] is T for every i in the set of names. > (For those who don't have Shoenfield, if I recall, where P is a > formula, P _x[i] is P[i|x], i.e, putting 'i' in for all free 'x'.) > > So if the set of names is empty: > > Then the function from the universe to the set of names is the empty > function. > > And x=x _x[i] for every i in the set of names. I'm not quite sure what is meant here. Can you use set notation (for 2-ary predicate) and 2-ary tuples instead? > > So the truth value of Ax x=x is T. I don't see how that could be arrived. If U = {}, all predicates including one for the 2-ary symbol '=' are also empty, which means no formula can be materially evaluated as true, in meta level. > > But then so is the truth value of Ax ~x=x. > > For ANY formula P with at most 'x' free, the truth value of Ax P is T. > > And that does make sense in this regard: since then for any formula P > with at most 'x' free, we have > > the truth value of Ex P is F > > which makes sense for an empty universe. > > MoeBlee > > > >
From: Nam Nguyen on 18 Jun 2010 02:32 Nam Nguyen wrote: > MoeBlee wrote: >> On Jun 17, 5:02 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >>> Nam Nguyen <namducngu...(a)shaw.ca> writes: >>>> So what's in you mind the definition of x=x being true be, model >>>> theoretically speaking? Can you _explicitly_ state the definition? >>> Just read some more Shoenfield. >> >> Wait, if 'x' is a variable, then, at least by pg. 19, there is no >> truth value for "x=x". > > That's actually not true, in closer inspection. What might make > one easily confused on the issue is he actually refering to 2 > different formulas: the original L in question, and the L(fancyA). > Then he said (pg. 19): "We shall now define a truth value fancyA(A) > for each closed formula in L(fancyA)"! > > So the closed formula A is in L(fancyA). So x=x is not in L(fancyA), > an extension of L, in the sense that x=x is already in L. > >> That is, Shoenfield, gives truth values only >> for closed formulas (sentences), and he bypasses defining >> 'satisfaction of a formula' (for formulas in general including open >> formulas) with a certain technique. Right? > > No. He actually defined A (of L) being true on pg. 18, and pg. 19, > where A is _either_ closed or open! [On pg. 19, this is in the > paragraph before the last lemma.] I forgot to mention: being true in a structure is being valid in the structure, which is having appropriate n-tuples _in_ an appropriate non-empty n-ary predicate. Since there are _no n-tuples_ _in_ an empty predicate and since there's no non-empty predicate in an empty U, no formula can be true in the degenerated structure where U is empty. In syntactical provability, proof and no-proof of A and ~A would follow nicely the pattern of binary boolean algebra of T and F, but _only_ when T is consistent. Similarly, being true and being false of a formula using set membership in a structure's predicate-sets would follow the same algebra _only_ in the typical case where U is NOT empty. When U is empty, logical reasoning using truth value of a formula would make as much sense as using provability in an inconsistent T. If all formulas are provable in an inconsistent T, then all formulas are false the structure of L(T) where its U is empty. >> Also, Aatu, I haven't been following this thread in every detail. That >> said, if I'm not mistaken there is one glitch that has to be addressed >> in allowing a version of Shoenfield with empty domains. >> >> In Shoenfield's definition of a structure, (when truly formalized) >> there is a function from the set of constants into the universe: >> >> "...for each constant e of L, e_fancyA is an individual of fancyA." >> >> I take that formally to mean there is a function f from the set of >> constants into the universe of fancyA (universe of fancy_A = | >> fancyA|): >> >> f(e) = e_fancyA >> and >> e_fancyA in |fancyA| >> >> So there must exist f such that >> >> f:set_of-constants -> |fancyA| >> >> But if the universe is empty while the set of constants is non-empty >> then there is no such function. >> >> So, I don't see how we can have an empty universe for a structure for >> a language that has constants. > > In such case, you just map any language constant C to the empty set {}, > to denote there's no (named) element in U to associate C with. > >> >> Meanwhile, let me see whether I understand the heart of the discussion >> correctly: >> >> The truth value of Ax x=x (where 'x' is a variable) is T iff >> >> the truth value of x=x _x[i] is T for every i in the set of names. >> (For those who don't have Shoenfield, if I recall, where P is a >> formula, P _x[i] is P[i|x], i.e, putting 'i' in for all free 'x'.) >> >> So if the set of names is empty: >> >> Then the function from the universe to the set of names is the empty >> function. >> >> And x=x _x[i] for every i in the set of names. > > I'm not quite sure what is meant here. Can you use set notation (for 2-ary > predicate) and 2-ary tuples instead? > >> >> So the truth value of Ax x=x is T. > > I don't see how that could be arrived. If U = {}, all predicates including > one for the 2-ary symbol '=' are also empty, which means no formula can be > materially evaluated as true, in meta level. > >> >> But then so is the truth value of Ax ~x=x. >> >> For ANY formula P with at most 'x' free, the truth value of Ax P is T. >> >> And that does make sense in this regard: since then for any formula P >> with at most 'x' free, we have >> >> the truth value of Ex P is F >> >> which makes sense for an empty universe. >> >> MoeBlee
From: Marshall on 18 Jun 2010 02:35
On Jun 17, 10:35 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > MoeBlee wrote: > > > So, I don't see how we can have an empty universe for a structure for > > a language that has constants. > > In such case, you just map any language constant C to the empty set {}, > to denote there's no (named) element in U to associate C with. Obviously that won't work. Constants identify individual members of the carrier set. If the carrier set is empty, there are no individual members to identify. Hence if the carrier set is empty, the model cannot be a model for a language with constants. (Exactly as Jesse said.) > > So the truth value of Ax x=x is T. > > I don't see how that could be arrived. If U = {}, all predicates including > one for the 2-ary symbol '=' are also empty, It's true that all predicates of the language are necessarily empty if U is empty. But this is irrelevant for this formula, because "Ax x=x" has no element of the signature in it. > which means no formula can be > materially evaluated as true, in meta level. No, it doesn't mean that. It means no *predicate* taken from the signature can evaluate to true, since all such predicates are empty. But that's only a proper subset of all formulas. Marshall |