Counterintuitions and the well-ordering theorem
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From: Herman Jurjus on 28 Oct 2009 06:53 Rupert wrote: > 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. Bad news, Bill. We just went from 50% to 33% in the popularity polls. -- Cheers, Haunted Herman
From: Aatu Koskensilta on 28 Oct 2009 06:58 Herman Jurjus <hjmotz(a)hetnet.nl> writes: > Aatu Koskensilta wrote: > >> Why should this be obvious? > > Who says it is? Naturally, when you said you find determinacy evident (or as evident as choice) I took you to mean that determinacy as usually understood is evident. It appears I was mistaken in this. > 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. As obliquely stated above, it appears that what you mean by determinacy is not what I -- and set theorists in general -- mean by determinacy. And as already said, if we are to understand determinacy in terms of some alternative conception of set, sequences, what have you, this conception needs be explained. (And, mainly owing to nothing but my laziness, and by now rather severe lack of sleep, I haven't digested your alternative determinacy interpretation as it's present in your previous posts to Daryl -- it is thus possible my incessant requests for further explanation and elucidation are unreasonable and petulant.) >> 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? Why does it surprise that I of all people should lack the relevant intuitions? >> (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? No. But as it happens, my flatmate is colour-blind. He is in many regards a rather singular individual. I don't think he has any mathematical or philosophical aspirations, though. -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Herman Jurjus on 28 Oct 2009 07:05 Aatu Koskensilta wrote: > 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? Perhaps later, some time. -- Cheers, Herman Jurjus
From: Daryl McCullough on 28 Oct 2009 07:10 Herman Jurjus says... >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'. But that isn't an accurate paraphrase. That's the reason I brought up rock/paper/scissors. I know it's a different *kind* of game, but it is a game of sorts, and it is definitely *not* the case that "the first player has no winning strategy" implies that the second player has a way to prevent player 1 from winning. Also, the paraphrase is clearly not equivalent in games involving chance. So the claim "Player 1 has no winning strategy" is only equivalent to "Player 2 has some way to prevent player 1 from winning" for certain kinds of games. What kinds of games are those? Don't you need to show that a particular game is one of those kinds of games? The most we can say without further analysis is that if player 1 doesn't have a winning strategy, then player 2 might luck into winning. I know you would say that there is no chance in this game, but there is nondeterminism in the *future* moves made by the other player. So the most you can say off the bat is that if player 1 has no winning strategy, then there is a strategy for player 2 that *might* win, provided that player 1 makes the wrong choices in the future. >The only remaining possibility is that perhaps this defense is not a >/winning/ strategy for player 2. How could that happen? Well, it's a cooperative game. For player 1 to win, player 2 must cooperate (whether intentionally or not), and for player 1 to *lose*, player 2 must cooperate (whether intentionally or not). No strategy by itself is necessarily a winning or losing strategy independent of what the other player does. >Yes, it's shaky. But is it /more/ shaky than what we get into our heads >when we try to convince ourselves of AC or (especially) the power set >axiom? Since I don't share your intuition about AD, they are less shaky to *me*. >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. Are you sure that your intuition doesn't rely on the fact that if white wins, he wins in a finite number of moves? >It's a bit like with the Jordan curve theorem: it's nice that we can >prove it, but had our definitions been such that it had come out as >false, we would only have concluded that our definitions needed >revision, not that the Jordan curve theorem is false. Yes, I agree completely in this case. I feel that if it is false, then it means we have defined "continuous curve" incorrectly. As a matter of fact, I think it is provably *false* for some natural ways of formulating continuity. -- Daryl McCullough Ithaca, NY
From: Herman Jurjus on 28 Oct 2009 08:10
Daryl McCullough wrote: > Herman Jurjus says... > >> 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'. > > But that isn't an accurate paraphrase. I agree that that's a shady part. And of course there's not much more i can say. ZFC+AD is inconsistent, so the burden is on me to analyze my intuitions further before saying anything more. > That's the reason I > brought up rock/paper/scissors. Ah. You think that that analogy is good, but i think it isn't. If it were, it should be piece-a-cake to come up with a counterexample of AD. But it isn't - it involves making very complex sets using AC, and (afaik) there are no counterexamples to AD where the winning set is ZFC-definable. -- Cheers, Herman Jurjus |