Is This really a Proof that Standard Math is Inconsistent?
in [Logic]
|
Prev: why finite-number versus infinite-number cannot be a primitive concept as point or line #319 ; Correcting Math
Next: When Is it Easier to Prove a Stronger Result?
From: Jesse F. Hughes on 25 Jan 2010 13:13 Andrew Usher <k_over_hbarc(a)yahoo.com> writes: > On Jan 21, 7:57 am, David C. Ullrich <ullr...(a)math.okstate.edu> wrote: > >> >Only in a vacuous sense. Mathematicians do assume 'P xor not P' >> >because it is true, that is, true in real, informal logic. The fact >> >that Goedel's theorem shows that it is not always so in any formal >> >system >> >> For heaven's sake, where did you get the idea that Godel's >> theorem says that "P xoe not P" is not always so in any >> formal system? > > It shows that it isn't always provable, which in the context of > rigorous proof means the same thing. "P xor not P" is provable. Perhaps you mean to say that it shows there are certain sentences P such that neither P nor NOT P are provable. It does not follow that P xor NOT P is not provable. -- Jesse F. Hughes "So there is some sense in which your work is more akin to a work of mathematics than a banana is." -- Jim Ferry encourages James S. Harris
From: Andrew Usher on 25 Jan 2010 21:27 On Jan 25, 10:21 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> Joshua Cranmer <Pidgeo...(a)verizon.invalid> writes: > >> > Ultimately, a proof by contradiction assumes that a contradiction cannot > >> > be proved in said formal system. You assumed that a contradiction exists > >> > in T, which renders invalid a proof by contradiction. > > Perhaps you will suggest that he really didn't mean to use "invalid" > here, but surely I'm not as clever as you are. I assumed that he used > the word "invalid" because he meant invalid. The context seems to indicate otherwise, as his sentence would be meaningless with you definition of 'invalid'. > (If I were really pedantic, I would ask you what you mean when you say > we're claiming that a proof is "true", by the way.) Is that really a definable term? True means true, in the universe of discourse (which for him, is standard mathematics). Andrew Usher
From: Andrew Usher on 25 Jan 2010 21:28 On Jan 25, 12:13 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > "P xor not P" is provable. Perhaps you mean to say that it shows > there are certain sentences P such that neither P nor NOT P are > provable. It does not follow that P xor NOT P is not provable. Wouldn't that prove, though, that the system is consistent? Andrew Usher
From: Jesse F. Hughes on 25 Jan 2010 21:38
Andrew Usher <k_over_hbarc(a)yahoo.com> writes: > On Jan 25, 12:13 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > >> "P xor not P" is provable. Perhaps you mean to say that it shows >> there are certain sentences P such that neither P nor NOT P are >> provable. It does not follow that P xor NOT P is not provable. > > Wouldn't that prove, though, that the system is consistent? Yes, if you want to be pedantic, I should have said: Goedel's theorem shows that, if PA is consistent, then there is a sentence P such that neither P nor NOT P is provable. The point remains. Your claim that P xor NOT P is not provable is simply wrong. PA proves every sentence P xor NOT P -- regardless of whether it is consistent or not. -- Jesse F. Hughes "As you can see, I am unanimous in my opinion." -- Anthony A. Aiya-Oba (Poeter/Philosopher) |