Counterintuitions and the well-ordering theorem
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From: Aatu Koskensilta on 12 Oct 2009 20:24 Rupert <rupertmccallum(a)yahoo.com> writes: > Some *consequences* of the existence of a well-ordering of the reals, > such as the Banach-Tarski paradox, are counter-intuitive. Well, I would argue that no-one not already deep into set theory, analysis, etc. has any intuitions about such matters as touched on in the Banach-Tarski theorem -- in particular, the usual explanations of the supposed counter-intuitiveness depend on the baffling idea that non-measurable sets corresponds to "cuttings" in some physical sense. That aside, I was wondering specifically about the claim that the well-ordering theorem itself is counter-intuitive, as alluded to in e.g. the famous quip The axiom of choice is obviously true, the well-ordering theorem obviously false -- and who can tell of Zorn's lemma? Just what intuitions are contradicted by the well-ordering theorem? -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Nam Nguyen on 13 Oct 2009 00:21 Aatu Koskensilta wrote: > Rupert <rupertmccallum(a)yahoo.com> writes: > >> Some *consequences* of the existence of a well-ordering of the reals, >> such as the Banach-Tarski paradox, are counter-intuitive. > > Well, I would argue that no-one not already deep into set theory, > analysis, etc. has any intuitions about such matters as touched on in > the Banach-Tarski theorem -- in particular, the usual explanations of > the supposed counter-intuitiveness depend on the baffling idea that > non-measurable sets corresponds to "cuttings" in some physical > sense. That aside, I was wondering specifically about the claim that the > well-ordering theorem itself is counter-intuitive, as alluded to in > e.g. the famous quip > > The axiom of choice is obviously true, the well-ordering theorem > obviously false -- and who can tell of Zorn's lemma? > > Just what intuitions are contradicted by the well-ordering theorem? > It's counter intuitive to assert either M(Pi) < M(e), or M(e) < M(Pi), where M(x) is the Major number of x and Pi and e are the 2 well known transcendentals (please refer to the thread "Ttranscendental Goldbach Conjecture"), since nobody would have a slightest intuition which one assertion is the case. Since the well-ordering of reals implies one assertion over the other is true, it's therefore counterintuitive. Put it differently, it's quite possible it's impossible to know which inequality would hold - in all models of reals. It's therefore counter- intuitive to talk about the well-ordering of the whole set of reals.
From: Herman Jurjus on 13 Oct 2009 05:32 Aatu Koskensilta wrote: > Rupert <rupertmccallum(a)yahoo.com> writes: > >> Some *consequences* of the existence of a well-ordering of the reals, >> such as the Banach-Tarski paradox, are counter-intuitive. > > Well, I would argue that no-one not already deep into set theory, > analysis, etc. has any intuitions about such matters as touched on in > the Banach-Tarski theorem -- in particular, the usual explanations of > the supposed counter-intuitiveness depend on the baffling idea that > non-measurable sets corresponds to "cuttings" in some physical > sense. That aside, I was wondering specifically about the claim that the > well-ordering theorem itself is counter-intuitive, as alluded to in > e.g. the famous quip > > The axiom of choice is obviously true, the well-ordering theorem > obviously false -- and who can tell of Zorn's lemma? > > Just what intuitions are contradicted by the well-ordering theorem? Imho, the quip tries to express something completely different. It's not an expression of mathematician's intuitions about sets. Mathematicians don't care about sets - they care about mathematics. And a general purpose foundation for mathematics should ideally not turn something into an indispensable truth that is, for mathematics, quite dispensable. Btw, personally i find Freyling's argument quite appealing. But i also think that AD and AC are both true, so you should better not take my opinions too seriously. (Not that there was any danger that you'd do that, anyway.) -- Cheers, Herman Jurjus
From: Jesse F. Hughes on 13 Oct 2009 08:07 Nam Nguyen <namducnguyen(a)shaw.ca> writes: > Aatu Koskensilta wrote: >> Rupert <rupertmccallum(a)yahoo.com> writes: >> >>> Some *consequences* of the existence of a well-ordering of the reals, >>> such as the Banach-Tarski paradox, are counter-intuitive. >> >> Well, I would argue that no-one not already deep into set theory, >> analysis, etc. has any intuitions about such matters as touched on in >> the Banach-Tarski theorem -- in particular, the usual explanations of >> the supposed counter-intuitiveness depend on the baffling idea that >> non-measurable sets corresponds to "cuttings" in some physical >> sense. That aside, I was wondering specifically about the claim that the >> well-ordering theorem itself is counter-intuitive, as alluded to in >> e.g. the famous quip >> >> The axiom of choice is obviously true, the well-ordering theorem >> obviously false -- and who can tell of Zorn's lemma? >> >> Just what intuitions are contradicted by the well-ordering theorem? >> > > It's counter intuitive to assert either M(Pi) < M(e), or M(e) < M(Pi), > where M(x) is the Major number of x and Pi and e are the 2 well known > transcendentals (please refer to the thread "Ttranscendental Goldbach > Conjecture"), since nobody would have a slightest intuition which one > assertion is the case. So, your problem is with the law of excluded middle? > Since the well-ordering of reals implies one assertion over the other > is true, it's therefore counterintuitive. That's not what I'd call counterintuitive. There are all sorts of things that satisfy this. There is a nickel in my drawer, where I can't see it. It is either heads up or tails up, but I don't have the slightest intuition which one is the case. Do you find that situation counterintuitive? > Put it differently, it's quite possible it's impossible to know > which inequality would hold - in all models of reals. It's therefore > counter- intuitive to talk about the well-ordering of the whole set > of reals. Yes, I think it would be counterintuitive to speak of "the" well-ordering, but surely people only speak of "a" well-ordering? -- Jesse F. Hughes "Why do the dirty villains always have to tie your hands *behind* ya?" "That's what makes them villains." --Adventures by Morse (old radio show)
From: Daryl McCullough on 13 Oct 2009 09:32
Aatu Koskensilta says... >Well, I would argue that no-one not already deep into set theory, >analysis, etc. has any intuitions about such matters as touched on in >the Banach-Tarski theorem -- in particular, the usual explanations of >the supposed counter-intuitiveness depend on the baffling idea that >non-measurable sets corresponds to "cuttings" in some physical >sense. I don't think what you are saying makes any sense. Bringing up non-measurable sets is not a way to explain the counter-intuitiveness of Banach-Tarski. It's a way of arguing that it *ISN'T* counter-intuitive (because people don't have good intuitions about non-measurable sets). The situation with Banach-Tarski to me is that (1) There is an informal, intuitively true claim, (something along the lines of: If you cut up a solid object into pieces, and you rearrange the pieces, you'll get a new object that has the same volume as the original. (2) There is an attempt to formalize the informal claim, by defining what a "piece" might be, what "rearrange" means, etc. (3) Then Banach-Tarski shows that the formal version is false. You can argue that the informal claim is true, and that the problem is that the formalization does not capture the informal notion of "piece". But it is completely wrong to say that the formalization depends on "baffling ideas" about non-measurable sets. The formalization doesn't *MENTION* non-measurable sets. Rather, the formalization just didn't specifically *RULE* *OUT* non-measurable sets. -- Daryl McCullough Ithaca, NY |