Counterintuitions and the well-ordering theorem
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From: WM on 13 Oct 2009 15:11 On 12 Okt., 20:16, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > It is often said that the well-ordering theorem, or the existence of a > well-ordering of the reals in particular, is counterintuitive. Alas, > I've never quite fathomed what intuitions are contradicted. Perhaps > someone with keener intuition into the mysteries of sets can shed some > light on this pressing matter? Certainly. Well-ordering is equivalent to AC. But AC ... "So, even though, for example, the Hausdorff-Banach-Tarski paradox has been called the most paradoxical result of the twentieth century, classical mathematicians have to convince themselves that it is natural, because it is a consequence of the Axiom of Choice, which classical mathematicians are determined to uphold, because the Axiom of Choice is required for important theorems which classical mathematicians regard as intuitively natural." [Henry Flynt: IS MATHEMATICS A SCIENTIFIC DISCIPLINE? (1994)] http://www.henryflynt.org/studies_sci/mathsci.html Regards, WM
From: WM on 13 Oct 2009 15:46 On 12 Okt., 21:46, Nam Nguyen <namducngu...(a)shaw.ca> wrote: > Aatu Koskensilta wrote: > > Nam Nguyen <namducngu...(a)shaw.ca> writes: > > >> Don't have a every details yet but it's possible the order be > >> relativistic over the (more absolute) existences of the reals, which > >> could give rise to paradoxes if we insist there's a "global" absolute > >> well order. Imho. > > > Whatever the merits of this, it appears to be a theoretical > > (philosophical?) doctrine or hypothesis, not an intuition contradicted > > by the well-ordering theorem. > > It's only as much as philosophical as the current mathematical reasoning > based on the assumed knowledge of the naturals is. If you already > preconceived the truth of well-order of the reals isn't philosophically > based It is based on an erroneous proof. Zeremlo's first proof of well ordering contains the sentence: "Wäre nun m' das erste Element von M', welches von dem entsprechenden Elemente m'' verschieden wäre, ..." meaning: If the gamma-set M' would differ from the gamma-set M'' by the element m' for the first time, ... [E. Zermelo: Beweis, daß jede Menge wohlgeordnet werden kann, Math. Ann. 59 (1904) 514-516] That implies: Zermelo's proof shows that the well-ordered set, e.g. R, can be brought into an unbroken linear sequence, called gamma-set M', were no element differs from the alternative sequence gamma-set M''. There is no first element missing in one of the sets, and every element has a precursor (otherwise it would not be sure whether the element without precursor was the first element that could possible be different in M' and M''). Hence, Zermelo proves that R can be put into an unbroken sequence, contradicting Cantor's proof. Regards, WM
From: WM on 13 Oct 2009 16:07 On 13 Okt., 02:24, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Just what intuitions are contradicted by the well-ordering theorem? Allegedly more numbers can be well-ordered than can be addressed or named. What else, however, is ordering but addressing or naming successively? > > -- > Aatu Koskensilta (aatu.koskensi...(a)uta.fi) > > "Wovon mann nicht sprechen kann, darüber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus There is no path to infinity, not even an endless one. [§ 123] It isn't just impossible "for us men" to run through the natural numbers one by one; it's impossible, it means nothing. [ ] you cant talk about all numbers, because there's no such thing as all numbers. [§ 124] Ludwig Wittgenstein, Philosophical Remarks Regards, WM
From: David C. Ullrich on 14 Oct 2009 10:17 On Mon, 12 Oct 2009 21:16:29 +0300, Aatu Koskensilta <aatu.koskensilta(a)uta.fi> wrote: > >It is often said that the well-ordering theorem, or the existence of a >well-ordering of the reals in particular, is counterintuitive. Alas, >I've never quite fathomed what intuitions are contradicted. Perhaps >someone with keener intuition into the mysteries of sets can shed some >light on this pressing matter? Come on. Surely it's clear that we need someone with a _less_ keen intuition to explain this. So I'll step in. It contradicts the inuitively clear fact that there is no well-ordering of R. Seriously. You're not going to get a clear _mathematical_ answer to your question. I don't think that anyone's suggested that the existence of a well-ordering of R is particularly counterintuitive _to_ someone who's actually studied set theory. But the quip about AC being obviously true while WOT is clearly false simply _is_ the way it seems to many people who haven't studied these things in any depth. Not that I can explain why that should be, but I can give evidence that it's so: One sees people who don't see the need for AC as an axiom since it's clearly true. One sees people writing books where they carefully state that this or that theorem requires AC, giving the impression that they want to make it clear what parts of the theory do and what parts do not depend on AC, but then in the same book they prove theorems that do depend on AC without acknowledging this, hence I conjecture without being aware of it. (Eg books on analysis treating, say, the Hahn-Banach theorem and also giving a "careful" proof that a countable union of countable sets is countable - a very smart analyst down the hall simply didn't believe me when I told him that that's not a theorem of ZF.) It does seem to me that AC is so intuitively clear to people that they use it all the time without realizing that that's what they're doing (I for one am always very nervous claiming that I have _not_ used AC anywhere in some argument). Otoh one sees the same people giving the impression that the existence of an uncountable well-ordered set requires AC. I'm not sure I've ever seen an _explicit_ statement to this effect, but that's certainly the impression one gets, especially when the author seems to be trying to avoid AC when it's not needed but uses it for this fact. It does seem to me that for whatever reason people _do_ use AC without even being aware of it, and that simply is not true of WOT. Because AC simply _is_ clear, while WOT is not. That's just observation of what people seem to do, hence it seems to me evidence of what their intuitions actually _are_ - WOT contradicts the intuition that R cannot be well-ordered, which is why we need AC, well known to have counterintuitive consequences, to prove it. David C. Ullrich "Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to." (John Jones, "My talk about Godel to the post-grads." in sci.logic.)
From: Herman Jurjus on 14 Oct 2009 11:52
David C. Ullrich wrote: > On Mon, 12 Oct 2009 21:16:29 +0300, Aatu Koskensilta > <aatu.koskensilta(a)uta.fi> wrote: > >> It is often said that the well-ordering theorem, or the existence of a >> well-ordering of the reals in particular, is counterintuitive. Alas, >> I've never quite fathomed what intuitions are contradicted. Perhaps >> someone with keener intuition into the mysteries of sets can shed some >> light on this pressing matter? > > Come on. Surely it's clear that we need someone with a _less_ > keen intuition to explain this. > > So I'll step in. It contradicts the inuitively clear fact that there > is no well-ordering of R. > > Seriously. You're not going to get a clear _mathematical_ > answer to your question. I don't think that anyone's suggested > that the existence of a well-ordering of R is particularly > counterintuitive _to_ someone who's actually studied set > theory. But the quip about AC being obviously true while > WOT is clearly false simply _is_ the way it seems to many > people who haven't studied these things in any depth. Not > that I can explain why that should be, but I can give > evidence that it's so: > > One sees people who don't see the need for AC as an > axiom since it's clearly true. > > One sees people writing books where they carefully > state that this or that theorem requires AC, giving the > impression that they want to make it clear what parts > of the theory do and what parts do not depend on AC, > but then in the same book they prove theorems that > do depend on AC without acknowledging this, hence > I conjecture without being aware of it. (Eg books > on analysis treating, say, the Hahn-Banach theorem > and also giving a "careful" proof that a countable > union of countable sets is countable - a very smart > analyst down the hall simply didn't believe me > when I told him that that's not a theorem of ZF.) It always strikes me that people talk about applications of countable choice or dependent choice as 'requiring AC'. I'd say that, when a result can be proved with just CC and DC, it makes much more sense to say that it does -not- require AC, even though perhaps it's not provable from ZF alone. CC is sooo much weaker. (And much less controversial, mathematically.) > It > does seem to me that AC is so intuitively clear to > people that they use it all the time without realizing that > that's what they're doing Do you remember any cases where full AC was really required, instead of just CC/DC? > (I for one am always very nervous > claiming that I have _not_ used AC anywhere in > some argument). > > Otoh one sees the same people giving the impression > that the existence of an uncountable well-ordered > set requires AC. I'm not sure I've ever seen an > _explicit_ statement to this effect, but that's > certainly the impression one gets, especially when > the author seems to be trying to avoid AC when it's > not needed but uses it for this fact. > > It does seem to me that for whatever reason people > _do_ use AC without even being aware of it, and > that simply is not true of WOT. Because AC simply > _is_ clear, while WOT is not. > > That's just observation of what people seem to do, > hence it seems to me evidence of what their intuitions > actually _are_ - WOT contradicts the intuition that > R cannot be well-ordered, which is why we need AC, > well known to have counterintuitive consequences, > to prove it. Nothing snipped; spot on! -- Cheers, Herman Jurjus |