Re: Supertask, superchicken, paradox
- From: "William Hughes" <wpihughes@xxxxxxxxxxx>
- Date: 23 Aug 2005 10:53:54 -0700
david petry wrote:
> In a recent thread, people were discussing "supertasks", and I
> haven't completely caught on to what's going on.
>
> Consider the following supertask.
This is basically the same 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
>
True, but totally irrelevant. There is no need to suppose that f
is continuous at infinity.
> So from this, we might say that in the limit as n goes to infinity,
> there is an egg in the nest.
We could, but it is a meaningless statement. What could "in the
limit as n goes to infinity, there is an egg in the nest" mean.
>
> 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.
>
Not quite. The statement is that, *at infinity* the nest is empty.
There is no attempt to use or define limits.
Up till now we have not defined "the number of eggs in the nest
at infinity". On possible solution is to declare the term
meaningless. Another is to say
a: if an egg is placed in the nest and not removed
it is in the nest at infinity
b: if an egg is removed from the nest and not subsequently
replaced, the egg is not in the nest at infinity.
a and b make sense (and more, any way of determining whether
an egg is in the nest at infinity that is not consistent with a
and b does not make sense). So if we are going to define the
number of eggs in the nest at infinity we have to use a and b.
This leads quickly to the conclusion that there are no eggs
in the nest at infinity. So the two possible aswers are
-the supertask does not make sense
-the nest is empty at infinity
- William Hughes
.
- References:
- Supertask, superchicken, paradox
- From: david petry
- Supertask, superchicken, paradox
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