The End of an Era - That of The Natural Numbers
in [Math]
|
From: Aatu Koskensilta on 16 Jun 2010 12:09 MoeBlee <jazzmobe(a)hotmail.com> writes: > 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. Sure, but we can just drop the requirement that the domain of a structure is nonempty. Nothing needs be changed in Shoenfield's definition of truth in a structure. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon man nicht sprechan kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: MoeBlee on 16 Jun 2010 12:20 On Jun 16, 11:09 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > MoeBlee <jazzm...(a)hotmail.com> writes: > > 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. > > Sure, but we can just drop the requirement that the domain of a > structure is nonempty. Nothing needs be changed in Shoenfield's > definition of truth in a structure. That well may be. But my question is specifically whether NAM agrees with the first two statements I mentioned. The point you mentioned would be a followup. I first want to be clear on Nam's view of the two statements I mentioned. MoeBlee
From: Nam Nguyen on 16 Jun 2010 21:27 MoeBlee wrote: > On Jun 15, 11:13 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > >> If you incorrectly assume a structure with an empty U isn't a structure, >> then of course you'd arrive such a wrong conclusion. > >>> A structure for a language is a way of consistently assigning "true" >>> or "false" to each closed formula in the language. > >> But such consistency could exist only in a structure with an U that's >> NOT empty. (As long as you don't acknowledge the case where U is empty >> your argument couldn't be successful here. Seriously: you got to _confront_ >> that case, whether or not you win or loose the debate.) > > I'd like to be completely clear on a certain point. > > Nam, do you agree or disagree with these statements: > > In Shoenfield, by definition, a structure for a language has for its > universe (domain of discourse) a non-empty set. Yes it's a fact that he chose to narrow his definition to restrict U to be non-empty. > 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. 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. And the subject of FOL reasoning would transcend each individual author's writing/posting, which means unless the written source was so bad that _really needs_ a revision, we could still see in the source some _common_ knowledge of FOL we could use to further inferences, or making arguments. Remember why we got here: Early on in the thread I had: >On a more serious note, my > > >>>>> In other word there's no absolute truth. > > only meant _within the context of FOL reasoning_ there's > no such thing as an absolute truth of a formula! (to which Marshall responded that x=x is such a formula). 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.
From: Nam Nguyen on 16 Jun 2010 22:16 Jesse F. Hughes wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> Jesse F. Hughes wrote: >>> stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: >>> >>>> Nam Nguyen says... >>>>> Marshall wrote: >>>>>> I thought that *you* were the one claiming that x=x is not true in >>>>>> all contexts. >>>>> I'm still claiming that. What have I just said that made you think >>>>> otherwise? >>>> To claim that a formula in a language L is not true in all contexts >>>> is to claim that there is a structure for L in which the formula is >>>> false, which is to claim that there is a structure for L in which >>>> the negation is true. There is no such structure. >>>> >>>> A structure for a language is a way of consistently assigning "true" >>>> or "false" to each closed formula in the language. >>> Given that Nam (allegedly) uses Shoenfield, I think you ought to stick >>> to Shoenfield's terminology. An open formula is neither true nor false, >>> but is instead either valid or invalid. >> So obviously you implied: >> >> (a) x=x is "neither true nor false". >> >>> The point remains, of course: x=x is valid, since it is true in every >>> interpretation of every structure. >> In "since it is true" it looked like by "it" you meant x=x. So apparently >> you meant: >> >> (b) x=x "is true". >> >> Why such a contradiction between (a) and (b)? >> > > No contradiction. Yes there is, because ... > > As a formula, x=x has no truth value. > > But each interpretation (or M-instance, in Shoenfield's terms) of x=x > has a truth value. in your (a) and (b) there isn't the phrase "each interpretation ... of x=x". Iow, you flatly said "x=x is neither true nor false" and then "x=x is true"! Read: whether or not it's an overlook, just admit a technical error when you make one and are asked about it.. > That is, for every structure and every assignment of > x to an element in the structure, the result is true. In Shoenfield's > terms, for every structure M, every M-instance of x=x has a truth value. > > A formula F is valid in M if every M-instance of F is true. It is valid > (simpliciter) if it is valid in every such M. Shoenfield even gives an > example of a valid formula, right there on p. 20. > > Know what it is? > > Yep. It's x=x. The formula x=x is valid for every structure (of its > language). That is, for every structure M, every M-instance of x=x is > true. > > You have read the first twenty pages of Shoenfield before declaring that > the era of natural numbers has ended, right? Sure. And on pg. 18 (read: it's before pg. 20) he assumed U is non-empty, when we're arguing whether or not x=x is false when U = {}, and you have been reminded about this a million times already. You really seem to have a reading comprehension problem!
From: Jesse F. Hughes on 16 Jun 2010 22:32
Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Jesse F. Hughes wrote: >> Nam Nguyen <namducnguyen(a)shaw.ca> writes: >> >>> Jesse F. Hughes wrote: >>>> stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: >>>> >>>>> Nam Nguyen says... >>>>>> Marshall wrote: >>>>>>> I thought that *you* were the one claiming that x=x is not true in >>>>>>> all contexts. >>>>>> I'm still claiming that. What have I just said that made you think >>>>>> otherwise? >>>>> To claim that a formula in a language L is not true in all contexts >>>>> is to claim that there is a structure for L in which the formula is >>>>> false, which is to claim that there is a structure for L in which >>>>> the negation is true. There is no such structure. >>>>> >>>>> A structure for a language is a way of consistently assigning "true" >>>>> or "false" to each closed formula in the language. >>>> Given that Nam (allegedly) uses Shoenfield, I think you ought to stick >>>> to Shoenfield's terminology. An open formula is neither true nor false, >>>> but is instead either valid or invalid. >>> So obviously you implied: >>> >>> (a) x=x is "neither true nor false". >>> >>>> The point remains, of course: x=x is valid, since it is true in every >>>> interpretation of every structure. >>> In "since it is true" it looked like by "it" you meant x=x. So apparently >>> you meant: >>> >>> (b) x=x "is true". >>> >>> Why such a contradiction between (a) and (b)? >>> >> >> No contradiction. > > Yes there is, because ... >> >> As a formula, x=x has no truth value. >> >> But each interpretation (or M-instance, in Shoenfield's terms) of x=x >> has a truth value. > > in your (a) and (b) there isn't the phrase "each interpretation ... of > x=x". Pardon me? It's sitting right up there. "It (x=x) is true in every interpretation of every structure." Now, I didn't use Shoenfield's terminology exactly, since I used the term "interpretation" rather than "M-instance", but you should have no doubt that I meant the exact same thing as Shoenfield's definition of validity. Also, to be fair, a suitably pedantic fella (Aatu?) might dislike my choice of preposition ("in" every interpretation), but surely this should not confuse a deep reader like Nam. Nonetheless, I'll say it again, in Shoenfield's terminology. x=x is valid because, for every structure M, every M-instance of x=x is true. This includes the empty structure[1]. Footnotes: [1] Although Shoenfield does not include the empty structure in his definition of structure, it is obvious that every M-instance of x=x is true when |M|={}. -- Jesse F. Hughes "It's not really winning if you don't get to where you want to go." -- An inspirational slogan from James S. Harris |