Cantor Confusion
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From: David Marcus on 21 Nov 2006 17:41 mueckenh(a)rz.fh-augsburg.de wrote: > Randy Poe schrieb: > > > > You never heard of countable uncountable models? > > > > I've never heard of something which is both countable > > (can be bijected to N) and uncountable (can not be bijected > > to N). That would seem to require both P and (not P) to > > be true for some proposition P. > > Indeed, that is my impression too. But ZFC cannot tolerate P and notP. > Therefore people have dispelled it. If ZFC is correct, then, according > to a theorem of Skolem, it must have a countable model. But the most > important theorem of ZFC proves the existence of uncountable sets. > Therefore the worshippers of transfinity proclaim the existence of a > set which is countable (as Skolem requires) "from outside" but does not > contain the bijection with N internally. Therefore "inside the model", > countability cannot be proved - and all is fine. Yes, all is fine. It appears you don't know what the word "model" means in mathematical logic. If you do know, please enlighten us by telling us what it means. -- David Marcus
From: David Marcus on 21 Nov 2006 18:16 Eckard Blumschein wrote: > On 11/9/2006 12:17 AM, Virgil wrote: > > > As infinitely many paths pass through each edge no path can "own" more > > than an infinitesimal share of that edge according to that "equal > > sharing" rule. And unless we are in something like Robinson's > > non-standard analysis, less than infinitesimal means zero. > > I wonder. I have to agree with Virgil. Is there something wrong? Just because you are wrong in your main points doesn't mean you can't sometimes be right when WM is wrong. It is hard to be always wrong. -- David Marcus
From: David Marcus on 21 Nov 2006 18:19 Eckard Blumschein wrote: > On 10/29/2006 10:22 PM, mueckenh(a)rz.fh-augsburg.de wrote: > > > Every set of natural numbers has a superset of natural numbers which is > > finite. Every! > > I am only aware of the unique natural numbers. If one imagines them > altogether like a set, then this bag may also be the only lonly one. WM means that every finite set of natural numbers is contained in a larger finite set of natural numbers. Since he believes all sets of natural numbers are finite, he then concludes that every set of natural numbers is contained in a finite set of natural numbers. This seems to be due to an allergy to the word "set". -- David Marcus
From: David Marcus on 21 Nov 2006 18:21 Eckard Blumschein wrote: > I would like to support Han because his expertise is much closer to > physics than the horizont of pure mathematicans. Do you really think so? > Maybe he even feels > equally reluctant to swallow |sign(0)|=0 like me. With aptly defined > reals I prefer |sign(0)|=1. Then, please tell us your definition of "reals" and of "sign". -- David Marcus
From: David Marcus on 21 Nov 2006 18:24
mueckenh(a)rz.fh-augsburg.de wrote: > Ralf Bader schrieb: > > > and Mueckenheim took that example for the main issue, ascribing > > some other weird opinions to Cantor on the way. Although, surprisingly, > > Mueckenheim's simplified proof of a restricted assertion is in principle > > correct, it is also telling that he seems to consider it worth to be > > published what actually is a mildly interesting exercise in basic point set > > topology. > > Unfortunately you missed the clue. The simplified version has only been > developed, because it can easily be extended to uncountable sets. > > I showed that the real plane even stays connected if an *uncountable* > set of points is left out. So Cantor's theorem is useless. It doesn't > show a difference between countable and uncountable, as Cantor might > have tried to suggest. So, it turns out that you were proving a different theorem than Ralph thought. That's reassuring. For a moment there, we thought you might have given a correct proof of something. Care to show us your proof that the plane is connected even if an uncountable number of points is deleted? -- David Marcus |