> Jesse F. Hughes wrote: > > Herman Jurjus <email@example.com> 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.
> 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