Counterintuitions and the well-ordering theorem
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From: John Jones on 14 Oct 2009 19:28 Aatu Koskensilta 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? > If mathematics is a universal language then it cannot syntactically present metatheoretical statements. In which case its formalism is restricted to the signs and symbols as they are presented on paper. In which case again, no intuitions come into mathematics.
From: Nam Nguyen on 15 Oct 2009 00:50 Jesse F. Hughes wrote: > 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? No. It's not LEM at all! That it's _impossible_to know the truth value of an arithmetic statement doesn't at all mean, say, Jesse or Nam is allowed to believe the statement's truth value is in between true and false. But it does mean Jesse could assert the statement be true and Nam false and it's _impossible_ to tell who's right or wrong. Which means the truth or falsehood of the statement is relativistic: if ones takes it as true, another one can _equally and logically_ take it as false. All the while LEM is still not broken. > >> 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. You're mistaken. The "All sort of things" cases don't have the word 'impossible'/'unknownability' in their descriptions. My cases do! Your side just never pays attention to the implication of those words (which I've cited numerous times). They do bring impact on the foundation of FOL reasoning. > There is a nickel in my drawer, where I can't see it. But can you or somebody else see it by walking up to the drawer and open it? So, can you see it, _in principle_? Of course you can! > 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? As just mentioned, of course you have knowledge about which one is the case _in principle_. So nothing is counterintuitive here. In brief, your analogy is close but not quite correct. Let me give a more precise analogy. Let's consider the statement: (1) In each of the _infinite_ number of universes there exists a planet with biological life during the planet's lifetime. Can you see the truth or falsehood of (1), even _in principle_? Of course (1) is either true or false and not in between. But there's a genuine _impossibility_ to know which truth value be the case. And, again, consequently the assertion that it be true or false is completely relativistic, depending which value one would _choose_. Closer to home, in FOL reasoning, there's an analogous situation. Whether or not it's true the last digit of a extraordinarily large prime number P is 1 is analogous to your "nickel in my drawer" case: we do know the truth _in principle_ because the proof of it can be done in _finite_ steps. But the arithmetic truth of say GC could be something else all together. It's either true or false of course (so LEM is not broken) but it could be impossible to know (since its "proof" could be infinite). So its truth would be relativistic, while LEM is not broken. > >> 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? > You missed the point: M(Pi) < M(e) could be relativistic in any model of reals. It's therefore counterintuitive to talk about any well-ordering of reals when it's _impossible_ to know if M(Pi) < M(e) is true or false. Put it differently, how could you talk about the order of numbers if it's _impossible_ to tell which of the 2 numbers is greater in the order ladder?
From: Jesse F. Hughes on 15 Oct 2009 07:58 Nam Nguyen <namducnguyen(a)shaw.ca> writes: >>> 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? >> > > You missed the point: M(Pi) < M(e) could be relativistic in any model > of reals. It's therefore counterintuitive to talk about any well-ordering > of reals when it's _impossible_ to know if M(Pi) < M(e) is true or false. > > Put it differently, how could you talk about the order of numbers if it's > _impossible_ to tell which of the 2 numbers is greater in the order > ladder? It's not impossible. It just depends on which well-ordering we're speaking of. Let M be any well-ordering of R, i.e., M:R -> |R|. Define a new well-ordering M' by M'(pi) = min{M(pi), M(e)} M'(e) = max{M(pi), M(e)} M'(x) = M(x) for all x distinct from pi and e. Now, it is perfectly clear that M' is a well-ordering of R and M'(pi) < M'(e). So, your claim that it is impossible to compare pi and e in a well-ordering of R is simply false. -- "Customers have come to SCO asking what they can do to respect and help protect the rights of the SCO intellectual property in Linux. SCO has created the Intellectual Property License for Linux in response to these customers needs." -- SCO responds to needs.
From: David C. Ullrich on 15 Oct 2009 11:25 On Wed, 14 Oct 2009 17:52:44 +0200, Herman Jurjus <hjmotz(a)hetnet.nl> wrote: >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? No. Yes, that would be much better for the point I'm making. >> (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!
From: Nam Nguyen on 16 Oct 2009 01:16
Jesse F. Hughes wrote: > Nam Nguyen <namducnguyen(a)shaw.ca> writes: > >> As just mentioned, of course you have knowledge about which one is the >> case _in principle_. So nothing is counterintuitive here. >> >> In brief, your analogy is close but not quite correct. Let me give a >> more precise analogy. Let's consider the statement: >> >> (1) In each of the _infinite_ number of universes there exists a planet >> with biological life during the planet's lifetime. >> >> Can you see the truth or falsehood of (1), even _in principle_? >> >> Of course (1) is either true or false and not in between. But there's >> a genuine _impossibility_ to know which truth value be the case. And, >> again, consequently the assertion that it be true or false is completely >> relativistic, depending which value one would _choose_. > > No, that clause beginning "consequently" does not follow. > > I'll assume, as you say, that either (1) is true or false, and we > don't know (and can't know) which. Right; although it should be reminded that there's a distinction between "don't know" and "can't [possibly] know": the later logically implies the former, but not the other way around! The case for (1) is the "can't [possibly] know" case; and that would make a difference. > It simply doesn't follow that the > truth value of (1) depends on whether I choose to believe it or not. If, given an underlying reasoning framework, a truth of a statement can't be possibly known then the truth is _undecidable_, notwithstanding the fact it's either true or false. And whence it's undecidable, one is *free to _decide_* which way it be! > Rather, either I choose correctly or I don't, but (1) is true or false > independently of my choice. Once something is undecidable, there's *no _correct_* choosing or deciding! > >> Closer to home, in FOL reasoning, there's an analogous situation. Whether >> or not it's true the last digit of a extraordinarily large prime number P >> is 1 is analogous to your "nickel in my drawer" case: we do know the truth >> _in principle_ because the proof of it can be done in _finite_ steps. >> >> But the arithmetic truth of say GC could be something else all together. >> It's either true or false of course (so LEM is not broken) but it could >> be impossible to know (since its "proof" could be infinite). So its truth >> would be relativistic, while LEM is not broken. > > Again, either GC is true of the natural numbers or it is not. Right. A formula is either true or false in a model of a T, just as the simultaneity of 2 events is either true or false in a frame of reference. But which frame are you talking about? Similarly, which "natural numbers" are you referring to? The one in which GC is true? Or the one in which cGC (hence ~GC) is true? > This fact doesn't depend on what we choose to believe. Which _undecidable_ fact are you referring to? The "natural numbers"-fact in which GC is true? Or the "natural numbers"-fact where it's false? > Your claims about relativism are simply wrong. Rather, you don't seem to understand how mathematical _impossibility_ is equated to mathematical relativity. |