Another AC anomaly?
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From: Herman Jurjus on 23 Nov 2009 18:02 A wrote: > On Nov 23, 6:22 am, Herman Jurjus <hjm...(a)hetnet.nl> wrote: >> Has anyone seen this before? >> >> http://possiblyphilosophy.wordpress.com/2008/09/22/guessing-the-resul... >> >> I'm not sure yet what to conclude from it; that AC is horribly wrong, or >> that WM is horribly right, or something else altogether. > Am I misunderstanding this? Look at the following paragraph: > > "The stategy you should adopt runs as follows. At 1/n hrs past 12 you > should be able to work out exactly which equivalence class the > completed sequence of heads and tails that will eventually unfold is > in. You have been told the result of all the previous tosses, and you > know there are only finitely many tosses left to go, so you know the > eventual completed sequence can only differ from what you know about > it at finitely many places. Given you know which equivalence class > you�re in, you just guess as if the representative of that equivalence > class was correct about the current guess. So at 1/n hrs past 12 you > just look at how the representative sequence says the coin will land > and guess accordingly." > > Apparently we do not choose a strategy before time begins to pass, but > rather, we have only ever chosen a strategy for playing the game once > some nonzero amount of time (1/n hours past 12) has passed. But after > any nonzero amount of time has passed, we have made infinitely many > coin tosses already, and there are only finitely many remaining coin > tosses; so one of two things is true: either > > A) we have already guessed infinitely many coin tosses incorrectly, > and hence there is no winning strategy for us, or > B) we have already guessed only finitely many coin tosses incorrectly, > in which case any strategy we choose will be a winning strategy, since > even if we guess all our remaining tosses wrong, we will still have > only guessed wrong a finite number of times. > > In other words, the author of the original article on Wordpress does > not seem to be telling us how to choose a winning strategy from the > beginning of this game, but rather, how to choose a winning strategy > once some finite amount of time has already passed; and in that case, > either there exists no winning strategy, or any strategy we choose is > a winning strategy. > > Perhaps I am misunderstanding something here, but this doesn't seem > like very novel stuff, and it doesn't seem to have anything to do with > the axiom of choice. That's also what I though, at first. But the strategy description as given leads to a real strategy only because/when you use AC as indicated. In general, not every next-move-prescription leads to a full strategy. Consider, for example, the next prescription: if all your guesses so far have been 'heads', then guess 'tails' next, in all other cases, guess 'heads' next. Although this determines a 'next move' in every situation, there's no overall strategy that fits this prescription, as is easy to see. (This particular example resembles the 'Yablo paradox', btw.) But by using AC in the way indicated, you do get a full strategy; that is: a function that assigns to every sequence of tosses one unique sequences of guesses, in such a way that every guess/move depends only on the previous events (either previous tosses or previous guesses). But as Butch Malahide has written, apparently this is already old hat in sci.math. -- Cheers, Herman Jurjus
From: Herman Jurjus on 23 Nov 2009 18:03 master1729 wrote: > Herman Jurjus wrote : > >> Has anyone seen this before? >> >> http://possiblyphilosophy.wordpress.com/2008/09/22/gue >> ssing-the-result-of-infinitely-many-coin-tosses/ >> >> I'm not sure yet what to conclude from it; that AC is >> horribly wrong, or >> that WM is horribly right, or something else >> altogether. > > well AC is horribly wrong. > > but i doubt that WM is horribly right. > > what is WM suppose to be right about ? That current mathematics too easily allows weird things involving infinity. (In this case the whole idea of a 'backwards infinite game'.) -- Cheers, Herman Jurjus
From: Butch Malahide on 23 Nov 2009 18:16 On Nov 23, 5:02 pm, Herman Jurjus <hjm...(a)hetnet.nl> wrote: > > But the strategy description as given leads to a real strategy only > because/when you use AC as indicated. In general, not every > next-move-prescription leads to a full strategy. > Consider, for example, the next prescription: > if all your guesses so far have been 'heads', then guess 'tails' next, > in all other cases, guess 'heads' next. > Although this determines a 'next move' in every situation, there's no > overall strategy that fits this prescription, as is easy to see. > (This particular example resembles the 'Yablo paradox', btw.) > > But by using AC in the way indicated, you do get a full strategy; that > is: a function that assigns to every sequence of tosses one unique > sequences of guesses, in such a way that every guess/move depends only > on the previous events (either previous tosses or previous guesses). In the present context, I think it is best to define a "strategy" as a function which depends only on the *opponent's* previous choices, i.e., on the previous tosses but *not* on the previous guesses.
From: master1729 on 23 Nov 2009 08:09 Herman Jurjus wrote : > master1729 wrote: > > Herman Jurjus wrote : > > > >> Has anyone seen this before? > >> > >> > http://possiblyphilosophy.wordpress.com/2008/09/22/gue > >> ssing-the-result-of-infinitely-many-coin-tosses/ > >> > >> I'm not sure yet what to conclude from it; that AC > is > >> horribly wrong, or > >> that WM is horribly right, or something else > >> altogether. > > > > well AC is horribly wrong. > > > > but i doubt that WM is horribly right. > > > > what is WM suppose to be right about ? > > That current mathematics too easily allows weird > things involving > infinity. (In this case the whole idea of a > 'backwards infinite game'.) > > -- > Cheers, > Herman Jurjus > WM is correct about that. Regards tommy1729
From: Butch Malahide on 23 Nov 2009 18:20
On Nov 23, 5:01 pm, Herman Jurjus <hjm...(a)hetnet.nl> wrote: > Butch Malahide wrote: > > On Nov 23, 5:22 am, Herman Jurjus <hjm...(a)hetnet.nl> wrote: > >> Has anyone seen this before? > > >>http://possiblyphilosophy.wordpress.com/2008/09/22/guessing-the-resul.... > > > In plain mathematical terms: for any set X, there is a function f:X^N- > >> X such that, for each sequence (x_1, x_2, ...} in X^N, the equality > > x_n = f(x_{n+1}, x_{n+2}, ...) holds for all but finitely many n. > > > This is old hat: see Problem 5348, American Mathematical Monthly, > > volume 72 (1965), p. 1136. (This has been discussed on sci.math in the > > past.) > > Good to know that! > When was that, approximately (I mean the sci.math discussion)? > And under what name was/is it known? It has probably come up more than once. I found some stuff by googling on "problem 5348". |