Counterintuitions and the well-ordering theorem
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From: Bill Taylor on 29 Oct 2009 21:15 > > choice seems to strike many as evident, both in the sense people > > explicitly state so > > 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? Another contributing factor being that almost all students are introduced to it, motivated to it, by thoughts of making an infinite sequence of choices. This convinces them of the common sense of Countable Choice, which is a far less questionable assumption than Wholesale Choice. Then the latter is slipped in more or less surreptitiously alongside the former. -- Wide-awake Willy
From: Bill Taylor on 29 Oct 2009 21:41 On Oct 28, 3:03 am, Herman Jurjus <hjm...(a)hetnet.nl> wrote: > that practically all > arguments crucially requiring (full) AC give the impression of being not > only not convincing at all, but of being particularly unenlightening, > like the invoking of a magic stick. Exactly so! Especially when it comes in the form of, "Let's well-order everything we've got, then etc etc" - that's when it *really* looks incredibly dodgy cheating! > AC being acceptable a.o. on the grounds that > you mention above: it's arithmetically conservative. Indeed, that is a powerful fact that suggests we might adopt it, though it also suggest it "isn't really saying anything" in some sense. Also, there is the point that *any* statement known to be independent of ZFC may as well be adopted as not, as anything proved by using it cannot be "wrong", or at least cannot be disproved. So AC, CH, AD, Suslin's hypothesis etc could all be adopted from time to time, even though some cannot be adopted simutaneously! -- Broad-minded Bill
From: Bill Taylor on 29 Oct 2009 21:47 Herman Jurjus writes: > > Actually, could you explain in what sense chess or checkers are > > games of complete information, and rock/paper/scissors is *not*? > > Now it's my turn to say 'OUCH!' Indeed! > Look, there may be any amount of valid reasons to be suspicious of AD, > or to dislike it, or whatever. But your objection is not one of them, imho. Quite so. Daryl's point is just plain wrong. More accurately, the term "complete information" is a standard term in Game Theory, and Daryl is using it wrongly. And as a standard term, any (finite) game with complete info automatically has either a) a winning strategy for player 1; XOR b) a winning straegy for player 2, XOR c) a drawing strategy for both. -- Board-gaming Bill
From: Bill Taylor on 29 Oct 2009 21:54 stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > I don't understand why you think that "all games are determined" is > intuitively true. You gave the answer yourself, in terms of infinite-depth quantifiers. > It's not *obvious* that chess or checkers has a > winning strategy; it's *provable*. It IS. (That is, a winning strategy or a drawing strategy for both.) This just plain OBVIOUS to any game player. It was obvious to me even before I started high school. >To prove it, you have to use the > fact that they are finite-length games Yes, the proof requires finitude, but the intuition does NOT. Again, consider your own i-d quantifiers! > What reason is there for believing that the principle > applies to games for which is not provable? I think you are asking - whose intuitions about infinite-depth games is more reliable. Is it the man in the street, or the hardened game-player, or the hardened math-logician? Obviously mileage will vary! -- Board-gaming Bill
From: Bill Taylor on 29 Oct 2009 22:07
Herman Jurjus <hjm...(a)hetnet.nl> pounces: > So we have two [equivalent] descriptions > > With one of these descriptions, a certain conclusion is not > self-evident, with the other it is ( [+ caveats] ) > 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 think this is a very perceptive observation, and a neat way of putting it. -- Beaming Bill |