Counterintuitions and the well-ordering theorem
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From: David C. Ullrich on 19 Oct 2009 08:11 On Sun, 18 Oct 2009 17:46:49 +0300, Aatu Koskensilta <aatu.koskensilta(a)uta.fi> wrote: >David C. Ullrich <dullrich(a)sprynet.com> writes: > >> Come on. Surely it's clear that we need someone with a _less_ keen >> intuition to explain this. > >So it appears. Regarding the evidence you provided for the evidence of >the axiom of choice (or at least dependent choice or countable choice), >well, there is a reason the axiom of choice is an axiom and the >well-ordering theorem a theorem. (Cantor did at one point think the >well-ordering principle was a "law of thought", but later came around >the view it's something that needs to be proven.) My favourite example >of a result where it's often difficult even for people with some >experience in such matters to spot the invocation of (countable) choice >is the theorem that omega_1 is regular. Ok, that's another exercise for me. What does "regular" mean here? >> Otoh one sees the same people giving the impression that the existence >> of an uncountable well-ordered set requires AC. > >This idea is not uncommon in news... 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: Rupert on 19 Oct 2009 08:16 On Oct 19, 10:11 pm, David C. Ullrich <dullr...(a)sprynet.com> wrote: > On Sun, 18 Oct 2009 17:46:49 +0300, Aatu Koskensilta > > <aatu.koskensi...(a)uta.fi> wrote: > >David C. Ullrich <dullr...(a)sprynet.com> writes: > > >> Come on. Surely it's clear that we need someone with a _less_ keen > >> intuition to explain this. > > >So it appears. Regarding the evidence you provided for the evidence of > >the axiom of choice (or at least dependent choice or countable choice), > >well, there is a reason the axiom of choice is an axiom and the > >well-ordering theorem a theorem. (Cantor did at one point think the > >well-ordering principle was a "law of thought", but later came around > >the view it's something that needs to be proven.) My favourite example > >of a result where it's often difficult even for people with some > >experience in such matters to spot the invocation of (countable) choice > >is the theorem that omega_1 is regular. > > Ok, that's another exercise for me. What does "regular" mean here? > A cardinal kappa is said to be regular if there does not exist an increasing well-ordered sequence of ordinals, the sequence being of length less than kappa, whose limit is kappa.
From: Herman Jurjus on 20 Oct 2009 09:04 Aatu Koskensilta wrote: > Herman Jurjus <hjmotz(a)hetnet.nl> writes: > >> Aatu Koskensilta wrote: >> >>> 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. > > Sure. What do you take the quip to (try to) express? > >> And a general purpose foundation for mathematics should ideally not >> turn something into an indispensable truth that is, for mathematics, >> quite dispensable. > > I'm not sure what you have in mind here. Well, it was my attempt to describe what the quip tried to express, but i think David Ullrich said it better: the quip itself -is- the best possible expression of the intuition. >> 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. > > What are your reasons for thinking AD true? That's a good question, and i'm still in the process of analyzing my intuitions in order to come up with something more tangible. I'm not sure whether anyone would be interested in more details, but since you asked for some, here comes. Simplistic subjective blather: If i forget for a moment that ZFC+AD is inconsistent, and i start with a clean sheet, so to say, and i give an honest account of what i mean with all the notions involved (N, set, game, sequence, etc.) then it becomes rather uncontroversial to me that either player 1 can win, or he can't, i.e. player 2 has some defense. The degree of 'reliability' that this has (for me) is not less than that of the power set axiom or AC (rather the opposite). Mathematical/pragmatic blather (short version): If AD fails, it seems to become pointless to accept a f.o.m. in which the LEM is an absolute truth for infinite sets. BTW, i only have these intuitions about AD as restricted to N^N. But AD on N^N is already in direct contradiction with the existence of a non-trivial ultra-filter on P(N), so this combination of intuitions is already weird enough. Just for what it's worth: Analyzing my intuitions has lead me so far in the direction of all kinds of semantics that explain the difference between potentially and actually infinite. The challenge (still open) is to come up with a system that has both types of infinity, and which makes AD true when the game-histories are allowed to be potentially infinite sequences, and false when they have to be actually infinite sequences. The weird thing is: i keep getting systems for which the 'potentially infinite part' also allows non-trivial ultrafilters on P(N) to exist. (I.e. the systems don't really resolve the clash of intuitions.) >> (Not that there was any danger that you'd do that, anyway.) > > I take nothing too seriously. Same here, and by all means let's keep it that way. -- Cheers, Herman Jurjus
From: David C. Ullrich on 20 Oct 2009 09:40 On Mon, 19 Oct 2009 05:16:18 -0700 (PDT), Rupert <rupertmccallum(a)yahoo.com> wrote: >On Oct 19, 10:11�pm, David C. Ullrich <dullr...(a)sprynet.com> wrote: >> On Sun, 18 Oct 2009 17:46:49 +0300, Aatu Koskensilta >> >> <aatu.koskensi...(a)uta.fi> wrote: >> >David C. Ullrich <dullr...(a)sprynet.com> writes: >> >> >> Come on. Surely it's clear that we need someone with a _less_ keen >> >> intuition to explain this. >> >> >So it appears. Regarding the evidence you provided for the evidence of >> >the axiom of choice (or at least dependent choice or countable choice), >> >well, there is a reason the axiom of choice is an axiom and the >> >well-ordering theorem a theorem. (Cantor did at one point think the >> >well-ordering principle was a "law of thought", but later came around >> >the view it's something that needs to be proven.) My favourite example >> >of a result where it's often difficult even for people with some >> >experience in such matters to spot the invocation of (countable) choice >> >is the theorem that omega_1 is regular. >> >> Ok, that's another exercise for me. What does "regular" mean here? >> > >A cardinal kappa is said to be regular if there does not exist an >increasing well-ordered sequence of ordinals, the sequence being of >length less than kappa, whose limit is kappa. Thanks - I recalled this a little later. 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: Daryl McCullough on 23 Oct 2009 11:39
Herman Jurjus says... >Simplistic subjective blather: >If i forget for a moment that ZFC+AD is inconsistent, and i start with a >clean sheet, so to say, and i give an honest account of what i mean with >all the notions involved (N, set, game, sequence, etc.) then it becomes >rather uncontroversial to me that either player 1 can win, or he >can't, i.e. player 2 has some defense. The degree of 'reliability' that >this has (for me) is not less than that of the power set axiom or AC >(rather the opposite). I know that you're not offering that as a mathematical argument, but I would like to understand your intuitions here. "Either player 1 has a winning strategy, or player 2 has a defense" needs a little more argument to be compelling, because of games like "Rock, paper, scissors" where no strategy is guaranteed to win. What's the argument against the possibility that (1) for every strategy for the first player, there is a defense for the second player, and (2) for every defense for the second player, there is a strategy for the first player that beats it. I don't see any reason to believe that this CAN'T be the case. So it's not clear to me that this has anything to do with the law of excluded middle. -- Daryl McCullough Ithaca, NY |