'When Are Relations Neither True Nor False?
in [Math]
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From: Daryl McCullough on 2 Apr 2010 06:23 Newberry says... >What I was getting at is how we know that the system is consistent. In the case of PA, it's because we know that everything it says about the natural numbers is true. Basically, the axioms of PA consist of: 1. Axioms that recursively define plus and times in terms of successor. 2. Axioms that say that zero is the smallest natural natural number, and that successor is 1-1. 3. The induction axioms, which basically say that every natural number is obtained from zero by repeatedly applying the successor function. -- Daryl McCullough Ithaca, NY
From: Jesse F. Hughes on 2 Apr 2010 06:16 Newberry <newberryxy(a)gmail.com> writes: > I do not know how it will turn out. I forgot who proved that the > square root of 2 was irrational and what his proof looked like. Maybe > your version is something concocted by the modern mathematicians who > take classical logic for granted. It was due to nameless Pythagorean. It was a geometric proof, rather than the more familiar algebraic proof, but I don't know the details. > Maybe it will turn invalid, maybe valid with some modifications or > added assumptions. Mind you the Greeks did not have the concept that > the vacuous sentences were true. The traditional syllogism > presupposes that the subject class is non- empty. I'll betcha that the mathematical proofs of, say, Euclid, do not follow the logical restrictions of Aristotle's categorical logic. But, again, I'm no expert on this by any stretch of the imagination. I would've thought that you had certain aims for your logic. I would've thought that, for instance, you would want that, if a set of sentences T entails P in your logic, then that same set of sentences entails P in classical logic. Roughly, that classical logic makes more things true, but doesn't make different things true. If so, of course, you'd have to drop the claim (recently made) that ~(Ex)(Px & Qx) -> (Ex)Px. Similarly, of course, I would expect that if your logic proves P, then so does classical logic. Right now, I'm not sure whether you've considered questions like this. If not, you prob'ly oughta. -- Jesse F. Hughes "When you try to kiss a girl, it's hard not to get spit on the girl." -- Quincy P. Hughes, age 3 (almost 4)
From: Daryl McCullough on 2 Apr 2010 06:24 Newberry says... > >On Apr 1, 5:59=A0am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> How do I know that Peano arithmetic is consistent? I know it the way I >> know any mathematical theorem I have personally proved. > >You proved PA consistent? It's easy to prove in ZF. -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 2 Apr 2010 10:00 Daryl McCullough wrote: > Newberry says... >> On Apr 1, 5:59=A0am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > >>> How do I know that Peano arithmetic is consistent? I know it the way I >>> know any mathematical theorem I have personally proved. >> You proved PA consistent? > > It's easy to prove in ZF. Is ZF _syntactically_ consistent?
From: Nam Nguyen on 2 Apr 2010 10:10
Daryl McCullough wrote: > Newberry says... > >> What I was getting at is how we know that the system is consistent. > > In the case of PA, it's because we know that everything it says > about the natural numbers is true. What is the natural numbers collectively? One in which "There are infinitely many examples of GC" is true? Or one in which "There are infinitely many counter examples of GC" is true? Are you sure you know what you're talking about, in talking about the "natural numbers"? > > Basically, the axioms of PA consist of: > 1. Axioms that recursively define plus and times in terms of successor. > 2. Axioms that say that zero is the smallest natural natural number, > and that successor is 1-1. Those are non-induction axioms and there are only finitely number of them and syntactically they don't mean anything, much less anything about "recursively". > 3. The induction axioms, which basically say that every natural > number is obtained from zero by repeatedly applying the successor > function. Again a FOL formula (or schema) doesn't have intrinsic meaning, or they could mean _anything_! |