'When Are Relations Neither True Nor False?
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From: Transfer Principle on 2 Apr 2010 21:53 On Apr 2, 5:32 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Nam Nguyen wrote: > > We shouldn't invent something that isn't compatible with rules of > > inference, and then "prove" things that the rules themselves > > can *not* prove. > To know the natural numbers then is to know the value of Un, which we > can not know. I can come up with similar "unknown" natural numbers, even easier than Nguyen just did. Let Un = n, if n is the smallest counterexample to GC = 0, if no such counterexample exists Or even: Let Un = 1, if a counterexample to GC exists = 0, if no such counterexample exists The so-called "crank" WM often came up with similar examples, such as Un = the googol'th digit of pi. Also, I can consider yet another example, f2(n), where n = Moser's number. Since PA, if consistent, must be incomplete, it's possible that we can never determine the value of some of these unknown numbers. Although not knowing the value of some of these unknown numbers doesn't in itself prove that PA is inconsistent, we must still remember Ed Nelson and his skepticism that PA is consistent. It's possible that we may be able to prove both Un = 1 and Un = 0 for one of these definitions of Un above. The proof of ~Con(PA) on which Ed Nelson is working involves some of these large numbers. I've said it before and I'll say it again -- every standard theorist believes it's _theoretically_ possible that PA is inconsistent, but no standard theorist believes it's _actually_ possible that PA is inconsistent.
From: Nam Nguyen on 2 Apr 2010 22:30 Transfer Principle wrote: > On Apr 2, 5:32 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Nam Nguyen wrote: >>> We shouldn't invent something that isn't compatible with rules of >>> inference, and then "prove" things that the rules themselves >>> can *not* prove. >> To know the natural numbers then is to know the value of Un, which we >> can not know. > > I can come up with similar "unknown" natural numbers, even easier > than Nguyen just did. > > Let Un = n, if n is the smallest counterexample to GC > = 0, if no such counterexample exists > > Or even: > > Let Un = 1, if a counterexample to GC exists > = 0, if no such counterexample exists The problem with your version of "Un" is should there be a counter example of GC (and that's still a possibility!), then your Un is known (in principle). My definition of Un would work - in all cases!
From: Newberry on 3 Apr 2010 00:05 On Apr 2, 3:23 am, stevendaryl3...(a)yahoo.com (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. > > 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. If it absolutely certain that PA is consistent why don't we formalize the reasoning? We have done it in English so we should be able to do it in a formal language. For example why don't we simply add PA is consistent as another axiom? > > -- > Daryl McCullough > Ithaca, NY
From: Newberry on 3 Apr 2010 00:16 On Apr 2, 3:16 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Newberry <newberr...(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. This is not an a priori goal. I have certain aims but this is not one of them. > 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. I have considered that my logic can prove different things. For example it can potentially prove ~(Ex)Pxm > > -- > 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: Newberry on 3 Apr 2010 00:18
On Apr 2, 7:00 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > 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? Such a proof in ZF that PA is consistent is obviously wothless. We had a long discussion about this a while ago. |