Re: Supertask, superchicken, paradox

"david petry" <david_lawrence_petry@xxxxxxxxx> writes:

> In a recent thread, people were discussing "supertasks", and I
> haven't completely caught on to what's going on.
>
> Consider the following supertask.
>
> Say we have a chicken which lays an egg, and when it does so,
> it immediately pushes any other eggs in its nest, out of the nest.
> Let's say it does this once every minute.
>
> Then define f(n) = 0 if there are no eggs in the nest at time n,
> and f(n) = 1 if there is at least one egg in the nest at time n.
>
> Then for all n >= 1, f(n) = 1, so in the limit as n -> oo, f(n) -> 1
>
> So from this, we might say that in the limit as n goes to infinity,
> there is an egg in the nest.
>
> However, the supertask experts seem to be saying that the
> right way to think about the supertask is to number the eggs
> 1,2,3 ... oo, and then to argue that for all n, egg_n has been
> removed from the nest, and hence in the limit as n goes to
> infinity, the nest is empty.
>
>
> So in the limit, is there one egg in the nest, or no eggs?

Seems to me that your no-egg analysis is spot-on.

Also seems to me that this is a good alteration of the problem. The
answer seems very counter-intuitive, since the chicken never pushed
out an egg without replacing it by another. How could it be empty
after the supertask? The answer "no eggs" sure seems surprising if
not just plain wrong. Nonetheless, I think that no eggs must be the
correct answer.

Possible consequences:

(1) Well, whaddya know? Our intuitions about infinity and supertasks
aren't so well trained.

(2) Grab a pitchfork and a torch and storm the Cantorian stronghold,
baby!

(3) Start talking about omelets with infinitely many eggs, and how
that requires eggs with an infinite number of yolks.

(4) Argue the problem away by the axiom x = {x} and thus the yolks
*are* the eggs.

Any others? (There's a fairly lame one I omitted: Nothing happens
*after* infinitely many minutes. Its cousin is: she can't keep
accelerating her rate without bound. Both of these just refuse to
play the game that David offers.)


--
"Now for once I might actually have an audience that realizes that
[my proof of Fermat's Last Theorem is correct], because you see,
they'll finally know what's in it for them--cold, hard cash."
--James Harris embarks on a new mathematical strategy.
.

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