Another AC anomaly?
in [Logic]
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From: David C. Ullrich on 24 Nov 2009 06:39 On Mon, 23 Nov 2009 15:03:25 +0100, Herman Jurjus <hjmotz(a)hetnet.nl> wrote: >David C. Ullrich wrote: >> On Mon, 23 Nov 2009 12:22:17 +0100, Herman Jurjus <hjmotz(a)hetnet.nl> >> wrote: >> >>> Has anyone seen this before? >>> >>> http://possiblyphilosophy.wordpress.com/2008/09/22/guessing-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. >>> >>> In short the story goes like this: >>> >>> A game is played, in which infinitely many coins are tossed, and there's >>> one player, who makes infinitely many guesses. Both are done over a >>> finite period of time. The tosses and guesses are not made faster and >>> faster, however, but slower and slower: at t = 1/n. There's no 'first' >>> move. >> >> In case I'm not the only one who couldn't figure out exactly what's >> going on from that paragraph, the description on the page seems >> more clear: >> >> "For each n > 0, at \frac{1}{n} hours past 12pm the following is going >> to happen: aware of the time, you are going to guess either heads or >> tails, and then I am going to flip a coin and show you the result so >> you can see if you are right or wrong. This process may have to be >> done at different speeds to fit it all in to the hour between 12pm and >> 1pm." >> >>> Claim: >>> There exists a strategy with which you're certain to guess all entries >>> correctly except for at most finitely many mistakes. Not 'certain' as in >>> 'probability is 100%', but absolutely certain. >>> >>> Reasoning: >>> On 2^w, consider the equivalence relation that makes x equivalent to y >>> when x(n) =/= y(n) for at most finitely many n. Next, using AC, create a >>> set S that contains precisely one element from every equivalence class. >>> Strategy: at every move, you already know the results of the previous >>> tosses, which is an infinite tail of some sequence in 2^w. Now take the >>> unique element from S associated to that tail, take the n'th element of >>> that sequence from S, and deliver that as your move. >>> After some thinking, you will see that with this strategy, you're indeed >>> certain to guess wrong at most finitely many times. >>> >>> Thanks, AC! Another nice mess you've gotten us into. >> >> I think the moral is not that AC leads to the weirdness but that >> this is a highly weird situation to begin with. At the time when >> we make any given guess we've already been told the result of >> infinitely many coin tosses... > >Yup - it's a game without a first move. So 'weird' is an accurate >qualification. Yet, it's not particularly difficult to give a >mathematical description of the situation, reason about it, and convince >ourselves that, mathematically, there's no problem with it. Ideally, it >should be much harder to make mathematical sense of a game like this. Why should this be hard? You wouldn't expect that being given infinitely much information would be very powerful? >That's why I included "WM is horribly right" as a possible moral. > >Apparently there's something wrong with backward supertasks (and not >with ordinary, 'forward' supertasks). But why should that be? David C. Ullrich "Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to." (John Jones, "My talk about Godel to the post-grads." in sci.logic.)
From: George Greene on 24 Nov 2009 09:59 On Nov 23, 9:03 am, Herman Jurjus <hjm...(a)hetnet.nl> wrote: > Yup - it's a game without a first move. So 'weird' is an accurate > qualification. Yet, it's not particularly difficult to give a > mathematical description of the situation, reason about it, and convince > ourselves that, mathematically, there's no problem with it. This is NOT true. The whole inference paradigm for standard classical FOL assumes that all rules of inference have FINITE number of premises. Even ONE attempt to make something depend on an infinite amount of prior information is a violation of the whole paradigm. This applies just as strongly to computing terms from arguments as it does to deriving conclusions from premises.
From: George Greene on 24 Nov 2009 10:04 On Nov 23, 9:48 am, Herman Jurjus <hjm...(a)hetnet.nl> wrote: > More conclusive (at least for me): you switch a light bulb on and off; > after infinitely many steps, is the light on or off? Well, there is more than one infinity. Ordinally, there are even successor ordinals after the first infinity. The reversed order (with no first element) can also have a last one. Basically, any infinity of steps that has a last element will have an answer to this question. Any infinite sequence that does not have a last element needs to get its "answer" from some NON-standard convention.
From: George Greene on 24 Nov 2009 10:09 > On Mon, 23 Nov 2009 15:03:25 +0100, Herman Jurjus <hjm...(a)hetnet.nl> > wrote: > >Yup - it's a game without a first move. So 'weird' is an accurate > >qualification. Yet, it's not particularly difficult to give a > >mathematical description of the situation, reason about it, and convince > >ourselves that, mathematically, there's no problem with it. Ideally, it > >should be much harder to make mathematical sense of a game like this. On Nov 24, 6:39 am, David C. Ullrich <dullr...(a)sprynet.com> wrote: > Why should this be hard? You wouldn't expect that being given > infinitely much information would be very powerful? Or even inadmissibly powerful; I mean, the definition of both a first-order language and the standard classical first-order inference paradigm PROHIBIT BOTH 1) functions from taking infinitely many arguments, AND 2) inference rules from taking infinitely many premises. You can legitimately work partially around this by using infinite sets as arguments IF they are finitarily definable. But when, as with AC, they are providing an infinite amount of information, then, yes, precisely as DCU says, you would EXPECT "problematic" situations to arise from the combination of "use of an infinite amount of information" in a context where that is AGAINST the "usual" policies.
From: Jesse F. Hughes on 24 Nov 2009 14:30
George Greene <greeneg(a)email.unc.edu> writes: > On Nov 23, 9:48 am, Herman Jurjus <hjm...(a)hetnet.nl> wrote: >> More conclusive (at least for me): you switch a light bulb on and off; >> after infinitely many steps, is the light on or off? > > Well, there is more than one infinity. > Ordinally, there are even successor ordinals after > the first infinity. The reversed order (with no first element) can > also have a last one. Basically, any infinity of steps that has > a last element will have an answer to this question. > Any infinite sequence that does not have a last element > needs to get its "answer" from some NON-standard convention. No, that's not enough. Suppose that the light begins "on", at each step, I toggle the state. It seems that you agree that after omega-many steps, we do not know whether the light is on or off. But if we do not know at omega whether the light is on or off, then surely we do not know whether it is on or off at omega + 1. Right? To put it differently, you claim "any infinity of steps that has a last element will have an answer to this question." w + 1 is an "infinity of steps" with a last element, but if we have an answer at w + 1, then we also have an answer at w. -- Jesse F. Hughes "Yes, I'm one of those arrogant people who tries to be quotable. There is actually at least one person who quotes me often." -- James Harris |