Counterintuitions and the well-ordering theorem
in [Logic]
|
Prev: Solutions manual to Entrepreneurship 1e Bygrave Zacharakis
Next: Solutions manual for Engineering Mechanics Statics (12th Edition) by Russell C. Hibbeler
From: Herman Jurjus on 28 Oct 2009 03:31 Aatu Koskensilta wrote: > Your stance, that both choice and determinacy are true, calls for some > further elucidation -- these principles are after all contradictory! I'm well aware of that. But i can't help my intuitions, you know. > The > usual set theoretic "intuition" is that determinacy holds for some > restricted class of sets Right. Making a counterexample to AD is terribly 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" > , the obvious choice for this class being > identified with L(R) very soon after the introduction of the axiom -- > which Mycielski didn't propose as a truth about sets, but precisely as a > principle that might be fruitfully studied, and hold, in context of some > subuniverse of sets -- and this is precisely what is borne out in the > study of large cardinals. Yes. The current way to deal with the matter is very sensible. 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.) -- Cheers, Herman Jurjus
From: Herman Jurjus on 28 Oct 2009 03:32 Daryl McCullough wrote: > Herman Jurjus says... > >> 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? > > You're right, that I substituted an alternative description that I > thought was more mathematically tractable to reason about. So I'll > try to stay to the first description: First player > selects a natural, then the second player selects a natural in > response, then back to the first player, etc. But I don't understand > how this description intuitively suggests that one player or the > other has a winning strategy. Well, if i had a /proof/ similar to yours, that would be very odd, of course. 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. How could that happen? Well, if after infinitely many moves it's not certain that player 2 has really won, for example because it's undetermined whether the game is won by player 1 or by player 2, but that is ruled out by assumption. 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? 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. 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. -- Cheers, Herman Jurjus
From: Herman Jurjus on 28 Oct 2009 03:40 Aatu Koskensilta wrote: > Herman Jurjus <hjmotz(a)hetnet.nl> writes: > >> Yah; it doesn't seem to make sense to continue much further with this. > > Indeed. I trust my continuing much further with this pleases or vexes > you no end. I was already wondering; for someone who takes nothing too seriously, you sure seem to spend much energy on this. -- Cheers, Herman Jurjus
From: Aatu Koskensilta on 28 Oct 2009 04:03 Herman Jurjus <hjmotz(a)hetnet.nl> writes: > Aatu Koskensilta wrote: > >> I trust my continuing much further with this pleases or vexes you no >> end. > > I was already wondering; for someone who takes nothing too seriously, > you sure seem to spend much energy on this. Why shouldn't we spend extraordinary amounts of energy on things we don't take too seriously? Surely it's a dismal, truly teeth-gnashing inducing idea, a revolting notion fit only to be rejected in a violent explosion of metaphorical vomit, that we only spend considerable length of time on stuff that we take very very seriously. That off my chest, I also find that I actually learn something of these electronic exchanges, coming out of them illuminated with new wisdom -- at least in case of exchanges with the likes of you, Daryl, Moeblee, David, ...; not necessarily anything of mathematical nature, but something of how people, myself included, react to this or that mode of argumentation, this or that way of putting this or that, this or that line of thought, this or that level of formality, gathering in the process valuable information about whether this or that way of putting an idea is generally intelligible, gaining for myself many a curious factoids about this or that English idiom and its use and abuse, what have you. Or this or that. Bits and pieces, follies and human insight. That's what I reap from these virtual encounters. It is also my hope that, in spite of my sometimes needlessly aggressive debating style, and peculiar and failed attempts at humour, those with whom I battle wits leave these Usenetical battlefields slightly improved, with a perspective on life just a whit expanded, or warped, from what it was before. It is customary in many newsgroups to include in otherwise off-topic posts a nugget of topicality. Here goes: Sigma-1 soundness implies Pi-2 soundness. But exactly on what conditions on the provability predicate? 'Tis a problem I leave you to ponder. -- 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 04:32
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? 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. (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. (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.) -- Aatu Koskensilta (aatu.koskensilta(a)uta.fi) "Wovon mann nicht sprechen kann, dar�ber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus |