Counterintuitions and the well-ordering theorem
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From: Bill Taylor on 29 Oct 2009 04:30 Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Bill is a sheep-shagging fence-sitter, Well now, I can hardly be both AT THE SAME TIME, can I!? Not unless we have some veeeeery strange sheep indeed! Kiwis don't mind sheep-shagging jokes, though I'm not too sure about the Welsh, who also get them. We particularly chuckle when Aussies try to make them, because we know they're just envious. Their sheep are miserable, scrawny, scruffy brownish critters that any respectable sheep-shagger wouldn't look twice at! Whereas OURS are lovely, plump, fluffy-white spotlessly clean lovable beasties. And we retort, "What do you call an Australian with a sheep under one arm and a goat under the other?" Answer:- "Bi-sexual!" > and a notorious proponent of an apparently > irremediably vaguely formulated form of "definitionalism" > when it comes to sets. No and yes. Even your much-admired and admirable Maddy has serious remarks about "definabilism", so it's hardly as muddy as you'd like to make out. And no, it's not irremediably vague, in fact we can make it (as I have done here before) fairly precise, and indeed have a paper in the pipeline about it. > This with all respect to Bill, who will no doubt take it all ingood humour. Of course! Whyever not? Just as you will take it in good humour when I allude to you as a luminally-challenged terminally depressive alcoholic from the far north with typically overly-terse unhelpful Scandinavian laconic replies to some questions; not but that you've been giving a LOT of extensive replies recently, thanks for that. And not that, technically speaking, you're really Scandinavian, but Fenno-Scandinavian. Though those who can tell the difference between Finns and other Scands, without names on them, are probably almost as rare as foreigners who can tell an Aussie & a Kiwi apart! Good-natured joshing is fine, it's when the ad hominems are used to further one's debating points that nastiness creeps in, as it does here, with others, from time to time.... -- Borealic Bill (NOT)
From: Butch Malahide on 29 Oct 2009 04:36 On Oct 29, 2:57 am, Bill Taylor <w.tay...(a)math.canterbury.ac.nz> wrote: > Now then, gentle Butch, to address your concerns:- > > On Oct 26, 8:50 pm, Butch Malahide <fred.gal...(a)gmail.com> wrote: > > >Can you explain why determinacy is intuitively clear when each move > >is choosing a natural number, but no longer clear when each move is > >choosing a countable ordinal? or a set of real numbers? > > This comment is too confusing for me - no countable ordinals or > reals have been mentioned up to now. However, this one... > > >What does your intuition say about the following game? First, White > >chooses a set X or real numbers; then Black chooses a realnumber x. > >Black wins if either X is empty or x is in X; White wins if X is > >nonempty and x is not in X. > > ...is a very good example indeed, and highlights a chink in > the alleged isomorphism between the consecutive and > simultaneous descriptions, and also the meaning of Choice itself. > It is a very good query indeed. > > Regarding the latter, Choice is only a logical/set-theoretical > problem when there are infinitely many to be made. However, > beginners in this area, having noted a(n alleged) problem with > Choice, often reduce it to even just a *single* choice; and ask, > "but how can you choose an element from *any* arbitrary (nonempty) > set anyway? - doesn't that mean being able to choose one from > *every* set - the main problem?" > > Well, we know they are wrong,though the confusion is understandable. > It's the old "any/every" problem again - being able to choose > an element from ANY nonempty set doesn't mean being able to choose > an element from EVERY nonempty set. The learner hasn't yet cottoned > on to the fact that *logic alone* allows one to choose an element > from ANY nonempty set - the epithet "nonempty" alone guarantees > that one can (almost magically, to the learner) do this. > Yes, it's weird. As von Neumann said: > "Young man, in math you don't get to > UNDERSTAND things, you just get used to them!" > .... > > What we have above, is the same principle applied to these games. > And thus highlighting a difference (at least an intuitive one) > between the consecutive and the simultaneous views of these games. > > Viewed consecutively, we have, at any moment, the single-choice > task, which is unproblematical. But viewed simultaneously, one > has (to be prepared) to make infinitely many choices all at once. > > Which is the "more intuitive" interpretation of what the game player > has to do? Obviously, mileage is going to vary on this one! > But kudos to Butch for winkling out this problem. > ... > > Another view, is to ask, what does it mean, intuitively, when > the game rules state, informally, that player 1 has to announce > a set of reals, for player 2 to choose a single member from; > in Butch's game? > > What is it, to "announce" a set. > Here is the heart of the (intuitive) problem. > > Surely it can't be allowed to just say nasty things like > "{2,3} if Goldbach and {4,5,6} if not". That's clearly cheating, > surely? OC that particular announcement could be met by player 2 > making a similarly conditional reply. But I expect it's easy enough > to make the cases sufficiently nasty to prevent this. > > So it must be that "announcing" a set of reals, in some way actually > *states* what are some particular elements in it. And as soon as this > is done, Butch's example evaporates. Because an "announcement" > that specifies some actual explicit reals, is going to specify > a first example in some way, and player 2 just goes with that. > > And thus we are back to considerations of Definitionalism, > which Aatu hates me talking about. But there it is. > .... > > Well Butch, there's my response. You may well think it a pile > of self-serving question-avoiding meretricious rubbish, > but I don't think so - I think there is something definite > to address there, insofar as there ever is when intuitive ideas > and language are used to introduce mathematical examples. > > Great query, though. > > -- Battling Bill My apologies, Bill. That was a pretty good answer. Not perfect, but no worse than my question was. You are right, that I need to cultivate the virtue of patience. I assume that, in due course, after considering the matter properly, you will award prizes to the co- winners of the POC tournament. :-) Perhaps you are waiting for one of them to die, so you only have to give one prize?
From: Daryl McCullough on 29 Oct 2009 08:00 Bill Taylor says... >What we have above, is the same principle applied to these games. >And thus highlighting a difference (at least an intuitive one) >between the consecutive and the simultaneous views of these games. Okay, I agree with that. But let's explore further. For simplicity, lets take a two-step game: Player 1 make a move m_1, then Player 2 make a move m_2. If Phi(m_1,m_2) then Player 1 wins, and otherwise Player 2 wins.in if ~Phi(m_1,m_2). So how would we formulate the claim: Player 2 can always win? It seems to me that there are two different formulations: 1. forall m_1, exists m_2, ~Phi(m_1, m_2) 2. exists g, forall m_1, ~Phi(m_1, g(m_1)) The first is the "consecutive" view, while the second is the "simultaneous" view. Assuming the axiom of choice, it doesn't make any difference. But if we *don't* want to assume the axiom of choice, then they are conceptually different. But now, let's go to *infinite* games. Without adopting the "simultaneous" view, how do you even *formulate* the claim that the second player can always win? If you formulate it as "there exists a strategy *function* for the second player such that forall strategy functions for the first player, player 2 can win", then you have used the "simultaneous" view. But if you reject the simultaneous view, how do you even formulate it, mathematically? To me, if you express the axiom of determinacy in terms of functions---Either there exists a winning strategy function for the first player, or there exists a winning strategy function for the second player---then you are implicitly adopting the simultaneous view, which only is sensible if you assume the axiom of choice, which contradicts the axiom of determinacy. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 29 Oct 2009 11:21 Butch Malahide says... >Can you explain why determinacy is intuitively clear when each move is >choosing a natural number, but no longer clear when each move is >choosing a countable ordinal? or a set of real numbers? > >What does your intuition say about the following game? First, White >chooses a set X or real numbers; then Black chooses a real number x. >Black wins if either X is empty or x is in X; White wins if X is >nonempty and x is not in X. I'm sure you will agree, Bill, that >neither player has a winning strategy. What gives? Okay, I've thought about it, and I realize that there is an answer to your question that makes AD seem a little less hypocritical (if that's the word for it). Rather than formulating "winning strategy" in terms of functions, we can formulate it in terms of sets of sequences. Define a "quasistrategy" Q to be a set of finite sequences such that whenever s is in Q, then Ax Ey s^[x,y] in Q (where s^[x,y] means s, followed by x, followed by y). The idea is that the elements of Q represent "good" positions for one player of the game when it is the other player's turn. To be good, it must be the case that no matter what your opponent does, it is possible for you to put the game back into a good position. If Q is a quasistrategy, then define the limit of Q to be the set of infinite sequences s such that every initial segment of s is in Q (or can be extended to an element of Q). In terms of "good" positions, s is in limit(Q) if s represents a game in which the player puts the game into a good position at every opportunity. Now, if W is a set of infinite sequences, then we say that Q is a winning quasistrategy for W for the first player if (1) limit(Q) is a subset of W, and (2) Ex [x] in Q In other words, the first player can put the game into a "good" position, and if he keeps it in a "good" position, then he will win (produce an infinite sequence in W). Q is a winning quasistrategy for W for the second player if (1) limit(Q) is disjoint from W, and (2) [] is in Q Now, we can formulate the axiom of determinacy in terms of quasi-strategies as: For every set W of infinite sequences, there is a quasi-strategy Q such that Q is a winning strategy for W for the first player or the second player. With this formulation, there is no restriction to W containing only infinite sequences of *naturals*. You can let the "elements" of the sequences be anything at all. For your case, we can let the elements of W be sets of reals. W = { [x_1, x_2, ...] such that x_2 is a singleton subset of x_1 } Then there is a winning quasi-strategy for the second player: Q = { [x_1, x_2, ..., x_n] | x_2 is a singleton subset of x_1 } That's kind of a boring notion of "strategy" in this case, but it fits the definition. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 29 Oct 2009 11:39
Bill Taylor says... >And not that, technically speaking, you're really Scandinavian, >but Fenno-Scandinavian. Though those who can tell the difference >between Finns and other Scands, without names on them, are probably >almost as rare as foreigners who can tell an Aussie & a Kiwi apart! Maybe it's not foolproof, but many of the Finns that I have known have eyes that look vaguely Asian. I don't know what the actual term is. There are also people of Irish ancestory who have that kind of eyes (maybe that's due to marauding Finnish Vikings?) -- Daryl McCullough Ithaca, NY |