Counterintuitions and the well-ordering theorem
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From: Herman Jurjus on 28 Oct 2009 04:58 Aatu Koskensilta wrote: > Herman Jurjus <hjmotz(a)hetnet.nl> writes: > >> But: either player 1 has a winning strategy or he hasn't. >> Now what does it mean for player 1 to not have a winning strategy? >> >> I'd say that amounts to 'player 2 has some way to prevent player 1 from >> winning'. >> >> The only remaining possibility is that perhaps this defense is not a >> /winning/ strategy for player 2. > > I think it's a good idea to explicitly introduce quantifiers, and the > quantifier switch in play, here. We are considering games where two > players choose integers in turn, the first player winning if the > resulting sequence is in a given subset A of the Baire space (the set of > infinite sequences of integers). Now, determinacy asserts that > > For all strategies for player 1, there is a strategy for player 2, such > that the resulting play does not end with player 1 winning. > > is equivalent to > > There is a strategy for player 2, such that for all strategies for > player 1, the resulting play ends with player 2 winning. > > Why should this be obvious? Who says it is? You seem to make the same mental move as Daryl: you rephrase the game as a simultaneous choice of strategies by the two players. With that description, the conclusion (determinacy) is indeed not obvious. This is what i wrote to Daryl about this: ] So we have two descriptions, which you claim to be equivalent. ] (I'm not so sure that the two are conceptually the same, but let's ] leave that, for now.) ] ] With one of these descriptions, a certain conclusion is not ] self-evident, with the other it is (at least for me; you don't ] agree with the latter). ] For me, the equivalence between the situations would lead me to ] accept the conclusion in the second case as well. You, on the other ] hand, seem to reject the conclusion in the first case, on the grounds ] that it is not self-evident in the second case. ] ] I cannot look inside your brain, but could it be that, so far, you ] simply /didn't/ intuitively evaluate the first description for ] yourself at all? I.e. that you just replaced it in your mind with the ] second description, and judged the situation based on that second ] description, while further totally ignoring the first description? ] ] [Because, in retrospect, that's what /i/ always did when thinking ] about AD, in the past.] > There are certainly games for which this > quantifier switch isn't valid (and I'm still bewildered by your > suggestion this is an instance of failure of the law of excluded > middle), so any argument or explanation for the evidence of AD must > depend in an essential way on the games being subsets of the Baire space > of descriptive set theory. But I can't see anything in your explanation > and arguments that wouldn't apply to just any infinite games, whence, in > part, my bafflement at your finding AD evident. Let me assure you that i'm just as baffled about it as you are. > (You have qualified your > stance on AD and AC in a post I haven't yet replied to, which I will > soon do, with an overly long and pointless rant, but this -- to my mind, > and apparently also to Butch's mind -- very important objection does not > hinge on the issues touched there.) > >> A propos chess and checkers: to me it /is/ immediately clear that >> either white has a winning strategy, or black has one that makes at >> least a draw, etc. It's nice that we can also prove it, but that's not >> really needed to see it's true. > > This is a nice example of divergent intuitions! My intuitions tell me > nothing whatever about the existence of winning strategies for chess or > checkers. Now /that/ is baffling. You of all people? > (And all I know about such matters is based on vague and hazy > recollections from game theory texts. I have played chess about five > times in my life, the plays consisting of my moving the pieces > essentially at random, the opponent declaring at some arbitrary, or so > it seemed to me, point that I'd lost. I'm also utterly tone deaf, and > have no aptitude for crossword puzzles; and mentally adding two > two-digit numbers takes about half a minute for me. I'm very proud of > and pleased with all these shortcomings, mainly because it irritates me > no end people think mathematicians should invariably be musically > talented, good at chess, etc.) You're not also color-blind, by any chance? -- Cheers, Herman Jurjus
From: Rupert on 28 Oct 2009 05:18 On Oct 28, 2:04 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Herman Jurjus says... > > >Might i humbly suggest that that is explainable by the fact that many > >people are confronted with choice in the earliest stages of their > >studies, and not with AD? > > >And what would have happened if ZFC+AD had not been inconsistent? Or if > >the inconsistency had not been discovered in the 1950s, but, say, in 2005? > > I don't understand why you think that "all games are determined" is > intuitively true. It's not *obvious* that chess or checkers has a > winning strategy; it's *provable*. To prove it, you have to use the > fact that they are finite-length games (actually, the possibility of > ties or cycles makes chess not precisely a finite sequential game, but > you can change the question to: is there some strategy that can guarantee > avoiding a loss?) What reason is there for believing that the principle > applies to games for which is not provable? > > -- > Daryl McCullough > Ithaca, NY I must say I agree. When I first read about AD I thought it was an interesting hypothesis, but I never had any feeling that it was intuitively plausible. On the other hand as soon as I encountered AC I was completely convinced that it was true.
From: Aatu Koskensilta on 28 Oct 2009 05:34 Rupert <rupertmccallum(a)yahoo.com> writes: > When I first read about AD I thought it was an interesting hypothesis, > but I never had any feeling that it was intuitively plausible. On the > other hand as soon as I encountered AC I was completely convinced that > it was true. Our agreement on these matters is most touching. Let's hug! Before we do that, I'll divulge the following piece of information about my personal history: when I first encountered determinacy, it immediately struck me as obviously false. I soon learned determinacy was intended to apply to sets that behave, sets that are in some intelligible manner built of basic, familiar and cozy base by means of (transfinitely iterated) operations that make (comfortable, familiar, cozy, mathematical) sense. Having learned this, AD (as restricted to such sets) appeared to me as a rather plausible hypothesis -- after all, since the sets are built up according to some sort of construction we can grok, on which we have a sort of "concrete" -- in so far as it makes sense to speak of concreteness in this context -- grasp on, it's not such a stretch to suppose this structuredness may be harnessed in constructing the required strategies. Here, by "plausible" I mean of course that I wouldn't have been terribly surprised if it turned out this is in fact the case. (And as it turns out, that /is/ in fact case, although I'm not particularly happy with the conceptual state of large large cardinal machinations.) In contrast, choice, replacement, powerset, etc. were all principles I would have thought very odd to call "plausible" -- they simply describe, in the form of assertions, what I usually do with sets that fall under my purview. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 28 Oct 2009 05:50 Herman Jurjus <hjmotz(a)hetnet.nl> writes: > If we accept both AD and AC as true, then we obviously have to give up > some of the other assumptions, either some of the other axioms of ZF, > or some of the inference rules of FOL, or some silent assumption that > we're not even aware of yet. Once we've identified and eliminated > it/them, who says that all consequences of ZF+AD will remain provable? If we follow this line of thought, the important question is, What are we talking about? In all my blather in this thread, I have spoken of sets in the world of sets as envisaged in the classic cumulative hierarchy story. If we can't take this background for granted, our first order of business is to try to work our what do we take for granted, what sort of sets are we talking about, and so on. As you note, it is not at all obvious whether standard results about determinacy, choice, large cardinals, are in the least relevant. So perhaps you will provide for us a fuller account of your conception of this world of sets where we find, so I've gathered, both potentially and actually infinite sequences, these sequences mingling in some sort of a synthesis of these and those intuitions? (I'm not going to edit that sentence any further, mainly because I'm suffering of severe sleep deprivation, but it really was not supposed to be insulting in any way. Colourful, perhaps, but not insulting.) -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 28 Oct 2009 06:03
Herman Jurjus <hjmotz(a)hetnet.nl> writes: > I'm well aware of that. But i can't help my intuitions, you know. Myself, I'm completely uninterested in most of my intuitions. On the vast majority of things, even in mathematics, I'm a completely uninformed layman, and consequently I don't see any good reason to pay any attention to my opinions, intuitions or ideas. > Right. Making a counterexample to AD is terribly difficult. Using choice it's very easy. (I'm sure you know the construction, so I won't bother sketching it here.) The natural question is thus: in what sense is it difficult? > You have to make a subset of N^N that is very badly behaved. So badly > behaved in fact that (for me) the point is reached where it's no > longer safe to claim that "for every sequence of naturals (being > produced step-by-step, in the course of time), the end result is > either in the set W or in its complement" "Safe" how? If we want to talk about these things in any interesting fashion, we must be clear on what we're saying. It is a classical triviality that a sequence is in W or in its complement. So, taking you at your word, we must have some non-classical conception in mind. It must be spelled out before any further progress is possible. > And my 'stance' is not meant as a criticism or rejection of that > practice, in any way. It's just a toy for me on rainy days, to see how > far it can lead us. (And the fact that nobody else seems to be > pursuing this, is exactly the reason why i do.) Well, bully for you! (I hate smileys, but have acquired a peculiar fondness for entirely uncalled for parenthetical explanations and clarifications, which fondness leads me to solemnly proclaim I didn't intend the "bully for you" sarcastically.) -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |