Is This really a Proof that Standard Math is Inconsistent?
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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)
From: Don Stockbauer on 29 Jan 2010 05:39
On Jan 25, 6:28 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > DonStockbauer<don.stockba...(a)gmail.com> writes: > > Actually, Hofstadter does spend a lot of time on Godel's results in > > GEB > > What is the relevance of this observation? As far as I can recall no one > here has claimed Hofstadter doesn't consider Gödel's theorems in some > detail in GEB. The wole quote is: Actually, Hofstadter does spend a lot of time on Godel's results in GEB, but the book's real value lies in getting people familiarized with the various memes which have formed/will refine the global brain. It's just being sneaky on my part. The more times the global brain is mentioned, the more times the global brain meme resides in the global brain, and this causes self-awareness of the global brain meme within the global brain, and self-awareness translates into higher intelligence for the global brain Don't you feel it all around you already, getting stronger every day, the global brain getting schmarter and schmarter? You can? Good. |