The End of an Era - That of The Natural Numbers
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From: Daryl McCullough on 15 Jun 2010 06:36 Nam Nguyen says... >What I asked you: > > > So, Marhsall, does the-thing-that-doesn't-equal-itself equal itself, > > mathematically speaking? The term "the-thing-that-doesn't-equal-itself" does not denote, so any question about it has no answer. Russell's approach to nondenoting terms was to rewrite them as follows: Psi(the x such that Phi(x)) ==> Exists x, Phi(x) & Psi(x) So for example, if Phi(x) == x is the present King of France, and Psi(x) == x is bald, then "The present King of France is bald" would be interpreted as "There exists an x such that x is the present King of France and x is bald" which is false. So if you are asking whether "The-thing-that-doesn't-equal-itself equals itself" is true, then under Russell's rewrite, it would become "There is an x such that x~=x and x=x" which is false. -- Daryl McCullough Ithaca, NY
From: Jesse F. Hughes on 15 Jun 2010 10:25 Aatu Koskensilta <aatu.koskensilta(a)uta.fi> writes: > stevendaryl3016(a)yahoo.com (Daryl McCullough) writes: > >> Yes, don't blame Marshall, blame me. > > I blame associate professor Chris Menzel. I blame a Rick Decker, a professor at Hamilton College. -- Scissors and string, scissors and string, When a man's single, he lives like a king. Needles and pins, needles and pins, When a man marries, his trouble begins. --- Mother Goose
From: Daryl McCullough on 15 Jun 2010 10:37 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. -- Daryl McCullough Ithaca, NY
From: Jesse F. Hughes on 15 Jun 2010 13:11 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. The point remains, of course: x=x is valid, since it is true in every interpretation of every structure. -- Jesse F. Hughes "Mathematicians don't fit in with a consistent view, unless you accept that to a strangely large extent they are acting under the influence of something very powerful, dark, and negative." -- James S. Harris
From: Nam Nguyen on 16 Jun 2010 00:13
Daryl McCullough wrote: > 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, Right. For instance, ~(1+1=0) isn't true in all structures. > which is to claim that there is a structure for L in which > the negation is true. Not necessarily. In structure with an empty U, no formula is true. > There is no such structure. 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.) |