Re: An infinite debate

Tony Orlow wrote:
Um, what if you start with 0, increment it at 1 minute before noon, then
again at a half minute before noon, then again at a third of a minute
before noon, etc, so that at time noon-1/n the number achieves a value
of n, for n in N? That way we have counted through all of the natural
numbers by noon, correct? What is the value of n at noon?

:)


David R Tribble wrote:
An infinite number of balls, uh, finite naturals in the vase?


Tony Orlow wrote:
We are talking about the value of the nth natural, as n->oo. It's that
simple.

Yes, it is quite simple. As you approach noon, you encounter more
and more naturals. By noon, you have encountered an infinite number
of naturals (all of them, in fact). But since there is no natural with
the value oo, n has no defined value at noon.

If we were adding numbered balls into a vase using the same schedule,
such that ball n is added at time 1/n minutes before noon, we would
end up with a vase full of an infinite number of balls, each labeled
with a unique natural number. There would be a ball for every natural
in the vase by noon.

Asking "what is the value of n at noon" is the same as asking "what is
the number of the ball inserted at noon?" Obviously, there is no ball
inserted at noon because we've used them all up before noon arrives.
So the answer is "none".

.

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