Obections to Cantor's Theory (Wikipedia article)
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From: Han de Bruijn on 20 Jul 2005 05:08 G. Frege wrote: > Look man: "Cantor's Theory" is of no interest any more these days. > Mathematics is concerned with modern _axiomatic_ set theory. And that is only of marginal interest to any other science, except mathematics itself (if it proclaims to be a science). Han de Bruijn
From: Han de Bruijn on 20 Jul 2005 05:23 stephen(a)nomail.com wrote: > It seems like a lot of the "anti-Cantorians" and other > mathematical nay-sayers tend to start with their conclusions, > and then try to work backward. The idea of starting > with a fixed set of well defined axioms and working > forward seems totally alien to them. Nah, nah. First comes the theorem. And then comes the proof. Han de Bruijn
From: David Kastrup on 20 Jul 2005 05:28 Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> writes: > David Kastrup wrote: > >> Oh good grief. Successor in interest to JSH, are we? > > JSH is not an anti-Cantorian. So this argument, again, doesn't make > sense and may only be useful for the purpose of insulting. JSH was the one who repeatedly threatened a day of reckoning when his opponents would be cast from the ranks of mathematicians. And that is exactly what Tony was doing here. I quote what you snipped: >>> I will have my web pages published before too long, so I am not >>> getting into a mosh pit with you again right now. Just be aware that >>> anti-Cantorians are sick of being called crackpots, and the day will >>> soon come when the crankiest Cantorians will eat their words, and >>> this rot will be extricated from mathematics. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: georgie on 20 Jul 2005 08:31 Stephen J. Herschkorn wrote: > Do anti-Cantorians accept that sum(i=1..infty, d_i / 10^i) "exists" > for each collection (d_i) of decimal digits (as i ranges over the > positive integers)? If so, how do they correctly justify the collection > of these real number are countable? What do you mean i=1..infty? I understand i=1, i=2, i=3, and so forth, but i=infty makes no sense. It looks to me like you are trying to "slip in" infty as a natural number. I have seen mathematicians along with smart people use such symbology when they mean to imply the sum of all terms as i takes on the values of all the natural numbers. I assume thats what you mean as well. (In this thread I think we should try to avoid ambiguities.) Ususally the limit of the sum is then discussed. That limit exists. I don't think many people would disagree. I don't see how limits of sums has anything to do with justifying the countability of "these real number."
From: David C. Ullrich on 20 Jul 2005 08:55
On Tue, 19 Jul 2005 17:19:52 +0100, Alec McKenzie <mckenzie(a)despammed.com> wrote: > David C. Ullrich <ullrich(a)math.okstate.edu> wrote: > >> Ok, here's another question. Suppose that we want to >> prove that A implies B. Suppose that we have an >> completely flawless proof that A implies C, and >> a completely flawless proof that C implies B. >> Does the union of those two proofs constitute >> a flawless proof that A implies B? > >Yes, I would say it does. > >> I imagine you'll say yes to that as well. But >> the proof of the theorem in question really >> does involve nothing more than statements >> which are true by definition and statements >> which follow from previous statements by >> "if A implies C and C implies B then A implies >> B" arguments. > >I would expect that to be the case for most direct proofs, if >not all. > >In the case of the proof of the theorem in question, we do not >already know for a fact that the conclusion is true; neither do >we know that it is false. If we did already know it was true >there would be little point in trying to find a flaw, regardless >of whether one might exist (there could be a flaw in the proof, >even if the conclusion is correct). But if we knew it to be >false, the flaw would have to be there even if we cannot find it. > >There are also proofs where we do know for a fact the conclusion >is false, even though there is no apparent flaw. The paradox of >the unexpected examination is an example of this, and I think >that if the conclusion in that case had been one, like Cantor, >where we had no way of knowing (apart from the proof) whether >the conclusion were true or false, the validity of the proof >would be almost universally accepted, and just as vigorously >defended. > >Equally, if the conclusion of Cantor's proof were known for a >fact to be false, it would be known as Cantor's paradox. Talking about whether the conclusion of the theorem is true or false is wrongheaded, it seems to me. The assertion is not that it's true in some absolute sense in the real world. What's asserted to be true in an absolute sense is that the conclusion follows from the axioms of set theory. (Much) more on this below. >My own feeling is that there exists the possibility, however >slight, that Cantor's conclusion is an obscure manifestation of >a paradox. We may or may not be getting somewhere. You've been suggesting that there may be a problem with the proof. That's simply not so, the proof _is_ correct. Yes, there have been incorrect proofs of other things published. Those were long complicated proofs. This proof is incredibly simple - one can easily hold the entire proof in one's head at one time, with plenty of room to spare. It's _correct_. Now, when you say that the theorem may be in some sense paradoxical that's not at all the same thing as saying that there is a problem with the proof. The explanation of what I mean by that may be clearer if we introduce four abbreviations, the first two of which are standard definitions: "ZF" = the standard axioms for set theory. "P is a theorem of ZF" means that P follows from the axioms of set theory by nothing but logic. "CT" = "There is no function mapping N onto P(N)." "NCT" = "There _is_ a function mapping N onto P(N)". Now, the actual theorem in question is this: (i) CT is a theorem of ZF. When you say that there may be problems with the proof of CT you're saying that (i) may be false, or at least that's what people are going to think you're saying. It is not true that (i) may be false - (i) is _true_, and anyone who doubts that it's true is misunderstanding something. Now, when you suggest there may be something paradoxical about CT I can think of at least two things you might mean. The first would indeed be paradoxical, but luckily we know for a fact that it's not so. The second may indeed be so, but luckily there's nothing paradoxical about it. Those two things you might mean, as far as I can see, are this: (ii) CT is a theorem of ZF, and slso CT is not a theorem of ZF. (iii) CT is a theorem of ZF, and NCT is also a theorem of ZF. If you're asserting (ii) you're simply wrong. (ii) is not possible, because CT _is_ a theorem of ZF. On the other hand, as far as anyone knows for certain, (iii) _could_ be true. Nobody thinks that's very likely, but nobody can prove it's not so. But if in fact (iii) is true there's nothing paradoxical about that fact - _if_ (iii) is true that simply says that ZF is inconsistent. If this turned out to be the case it would be a big deal, meaning that we'd have a lot of revising to do. But if (iii) is true that says _nothing_ about the _fact_ that CT _is_ a theorem of ZF. ************************ David C. Ullrich |