Another AC anomaly?
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From: master1729 on 24 Nov 2009 05:11 > 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 > -- James Harris indeed. and that one person who quotes JSH is JFH. tommy1729
From: Jesse F. Hughes on 24 Nov 2009 15:27 master1729 <tommy1729(a)gmail.com> writes: >> "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 >> -- James Harris > > indeed. > > and that one person who quotes JSH is JFH. You know what makes a good joke a great joke? When someone takes the time to explain it. -- Jesse F. Hughes "We will run this with the same kind of openness that we've run Windows." Steve Ballmer, speaking about MS's new ".Net" project.
From: Daryl McCullough on 24 Nov 2009 22:52 Jesse F. Hughes says... >Let's use the usual trick: For n in w, step n occurs at t - 1/n. >Let's also let step w + n occur at t + 1 - 1/n (for n > 0), so step=20 >w + 1 occurs at time t. Thus, we have actions occurring at > >t - 1, t - 1/2, t - 1/3, ... t, t + 1/2, t + 2/3, .... > >Seems to me if we're willing to grant the original supertask, then >there's nothing less plausible about the new supertask. > >At t - 1, the bulb is on. Immediately after t - 1 (and up to and >including t - 1/2), the bulb is off. Immediately after t - 1/2, it's >on again, and so on. > >Now, according to George, we should have no trouble figuring out what >happens immediately after time t, if I read him right. But in the >original task (when nothing was done *at* t), we didn't know the state >of the bulb at t. I don't see how we know the state of the bulb >immediately after t either. It's not specified by the "theory" of light switches. What we are assuming about light switches is that if it is switched on at time t_1, and it is not switched off between t_1 and t_2, then it is still on at time t_2. If it is switched off at time t_1, and is not switched on between t_1 and t_2, then it is still off at time t_2. Now, we add to this the assumption that the light switch is turned on at times t - 1, t - 1/3, t - 1/5, ..., and that is switched off at times t - 1/2, t - 1/4, ... So we have a bunch of statements about what the state of the light is at various times, and those statements don't determine what the state of the light is at time t. -- Daryl McCullough Ithaca, NY
From: T.H. Ray on 24 Nov 2009 21:34 Herman Jurjus wrote > Jesse F. Hughes wrote: > > Herman Jurjus <hjmotz(a)hetnet.nl> writes: > > > >> Apparently there's something wrong with backward > supertasks (and not > >> with ordinary, 'forward' supertasks). But why > should that be? > > > > Well, I'm not at all sure that there's no problem > with forward > > supertasks. Surely, it is not difficult to come up > with a > > problematic case. > > > > For instance, take our favorite example: at each > time t - 1/n, place > > balls 10(n-1) to 10n - 1 in a vase and then remove > ball n. At the end > > the vase is empty. > > > > Now alter the situation slightly. At each step, > again place 10 balls > > into the vase and then remove one ball, but remove > the ball > > *randomly*. At the end, the vase may contain any > number of balls. > > This strikes me as suitably counterintuitive to say > that the forward > > supertask has something wrong with it. Or, > perhaps, with my > > intuitions. > > More conclusive (at least for me): you switch a light > bulb on and off; > after infinitely many steps, is the light on or off? > (Or: you put one > and the same ball in the vase, out of the vase, in > the vase, out of the > vase, etc. What's in the vase/where's the ball after > infinitely many steps?) > If you can't trust supertask-reasoning in this case, > why should you > trust it in other, seemingly less problematic cases? > Infinity isn't a number. Infinitely many steps have no context in this problem. The problem description fixes the domain at countably infinite; however, even that term has no meaning in the absence of a higher cardinality. One has to mean, in this context, an arbitrarily large number of steps. In order to answer the question, though, one requires infinite information. That is, in an arbitrarily long string of 0s and 1s representing the state of the lamp at a discrete moment, one must be able to identify a segment of a certain magnitude and inspect the endpoint for the state at that singular moment. A little reflection informs us that information is not discrete--even though moments of time are countable; i.e., however short or long the segment, the initial condition (off or on) determines the final state (odd or even number). If there is no final state, there is no certain answer. The deep implications of discretely countable moments containing infinite information have led some researchers (Lev Goldfarb for one, I for another--though by very different routes) to the published conclusion that time and information are identical. A geometric, or pre- geometric approach, as she would say, leads Fotini Markopoulou to a theory that _only_ time, and not space, exists. In any case, though, the axiom of choice has no relevance to a question in which time plays the central role, because even though one may choose arbitrary initial conditions, such a choice compels no information that lies outside that model and which imposes a non-arbitrary moment on both our physical and mathematical experience, no matter whether we deem such a moment arbitrarily short or arbitrarily long. Tom > For the sake of clarity: I'm not endorsing or > advocating the > backward-supertask paradox as extremely important. > Just wanted to share > it here, because I had never heard of it, and thought > perhaps others > also hadn't. > > -- > Cheers, > Herman Jurjus
From: LauLuna on 25 Nov 2009 15:49
On Nov 23, 12:22 pm, 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. > > 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. > > 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. > > -- > Cheers, > Herman Jurjus In fact, you don't need the axiom of choice. At any 1/n hour past 12pm it is already determinate what equivalence class the eventual sequence is in. Since you know what the previous results are and there are only finitely many outstanding results, you can complete the sequence at random and take it as your representative. Which means that from any 1/n hr past 12pm on you can guess at random. That is, you can make all your guesses at random and still be certain to guess wrong only finitely many times. I'd say the paradox arises from the assumption that an infinite sequence of past events is terminated. So it seems akin to some versions of Benardete's and Yablo's paradoxes. The following is a version of Benardete's. Consider an infinite past with a gong peal occurring at each day and a hearer being deafened by it iff no previous gong peal has deafened it. The hearer must be deaf from eternity, which, paradoxically, implies that no gong peal deafens it. The problem is that at any day it is already determinate whether the hearer is deaf or not and in such a way that no event at no day can contribute to the fact. Similarly, what the equivalence class of the eventual sequence is, is determinate at any 1/n, no matter how big n is; hence no toss at no 1/ n contributes to the fact. Of course, this is paradoxical. Regards. |