Counterintuitions and the well-ordering theorem
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From: Herman Jurjus on 28 Oct 2009 09:40 Herman Jurjus wrote: > Daryl McCullough wrote: >> Herman Jurjus says... >>> 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. >> >> It's a completely airtight argument that the *lack* of a strategy >> for one player does not imply (without additional assumptions) the >> existence of a strategy for the other player. What are those >> additional assumptions? > > No hidden moves (where this includes 'adjourned' moves and simultaneous > moves), no chance-moves. Allow me to rephrase that: games with consecutive moves by both players (and no dice), like in a board game. And the 'extra intuition' that leads to accepting AD seems to amount to a form of 'extreme realism' in imagining that you can actually perform such infinite games. (I have no better way to express it, at this moment.) -- Cheers, Herman Jurjus
From: Daryl McCullough on 28 Oct 2009 18:19 In article <hc9hjl$45m$1(a)news.eternal-september.org>, Herman Jurjus says... > >Herman Jurjus wrote: >> Daryl McCullough wrote: >>> It's a completely airtight argument that the *lack* of a strategy >>> for one player does not imply (without additional assumptions) the >>> existence of a strategy for the other player. What are those >>> additional assumptions? >> >> No hidden moves (where this includes 'adjourned' moves and simultaneous >> moves), no chance-moves. > >Allow me to rephrase that: games with consecutive moves by both players >(and no dice), like in a board game. But what is the intuition behind believing that those conditions should be sufficient? >And the 'extra intuition' that leads to accepting AD seems to amount to >a form of 'extreme realism' in imagining that you can actually perform >such infinite games. (I have no better way to express it, at this moment.) I don't see how this extreme realism is relevant here. The question is not whether the game can be carried out; let's assume that the first move takes 1 second, the second move 1/2 second, etc., so the whole game is over in 2 seconds (and we know who won). So given an arbitrary pair of strategies f and g, they can easily figure out whether f beats g or not (when the first player follows f and the second player follows g). Furthermore, let's assume that given any strategy f, they can find a strategy g that beats f (if one exists), and given a strategy g, they can find a strategy f that beats g (if one exists). My problem is not imagining that such godlike beings might exist, but I don't see, even granting such godlike powers, that there is any reason to believe that one or the other will come up with a winning strategy. If there is a winning strategy, we can assume that they will find it, but why should there be such a strategy? Once again, I ask you to consider how such godlike beings would find the winning strategy for a finite game such as chess: Define recursively a sequence W_i of sets of positions as follows: W_0 = the set of all positions that are automatic wins for the first player (in chess, the check-mated positions for black). For n odd and greater than 0, W_n = the set of all positions such that any move by the second player results in a position in W_{n-1}. For n even and greater than 0, W_n = the set of all positions such that there exists a move by the first player resulting in a position in W_{n-1}. Having generated such a sequence, we can say: If the starting position is in some W_i, then the first player has a winning strategy: make any move so that the resulting position is in W_{i-1}. If the starting position is not in any of the W_i, then the second player has a winning strategy: make any move so that the resulting position is *not* in any of the W_i. Without going through such a argument in favor of "one player or the other has a winning strategy", I don't see why you would believe it. -- Daryl McCullough Ithaca, NY
From: Herman Jurjus on 28 Oct 2009 19:02 Daryl McCullough wrote: > In article <hc9hjl$45m$1(a)news.eternal-september.org>, Herman Jurjus says... >> Herman Jurjus wrote: >>> Daryl McCullough wrote: > >>>> It's a completely airtight argument that the *lack* of a strategy >>>> for one player does not imply (without additional assumptions) the >>>> existence of a strategy for the other player. What are those >>>> additional assumptions? >>> No hidden moves (where this includes 'adjourned' moves and simultaneous >>> moves), no chance-moves. >> Allow me to rephrase that: games with consecutive moves by both players >> (and no dice), like in a board game. > > But what is the intuition behind believing that those conditions > should be sufficient? [...] > Without going through such a argument in favor of "one player or > the other has a winning strategy", I don't see why you would believe > it. I can't say it better, at this stage, than i already did. Sorry. (Can you /explain/ /why/ you think the Jordan curve should definitely be true? I can't.) -- Cheers, Herman Jurjus
From: Bill Taylor on 29 Oct 2009 03:18 On Oct 28, 5:30 am, Butch Malahide <fred.gal...(a)gmail.com> wrote: > Actually, my comment was directed to Bill Taylor, who > is going to ignore it because he doesn't have a good answer. Oi oi oi! Temper, temper. Patience my dear boy. I am often slow to respond in these threads, because (1) I like to take a printout of the daily posts home, so as to peruse it more leisurely there, and (2) I don't like dashing into print before having time to consider the matter properly, and allow immediate intemperate would-be retorts time to simmer down. Unlike some here. So may of my articles will thus re-cover ground that has more recently been addressed; annnoying for some, but OTOH by delaying, I also reduce the chance of this happening. However, to get back to your query, I have a response, and will write it out right after this admonition. :) -- Brimming-over Bill
From: Bill Taylor on 29 Oct 2009 03:57
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 |