Re: An infinite debate

RLG wrote:
"Tony Orlow" <tony@xxxxxxxxxxxxx> wrote in message news:455388d9@xxxxxxxxxxxxxxxxxxxxxx
David Marcus 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?

This is a great question and I think the answer to it depends on what is meant by `increment' and what is meant by the value of `n' at noon. If by `n' we mean a variable name for a number stored in the registry of some kind of ideal computer then I don't think the value of `n' at noon is defined. But what if by `n' we mean the number of hash marks that could be made in a unit interval by noon? That is at 1 minute before noon a hash mark is made at the location -1 on the real line, at 1/2 minute before noon another hash mark is made at -1/2 on the real line, at 1/3 of a minute before noon yet another hash mark is made at -1/3 on the real line, and so on. At noon the total number of hash marks in the interval [-1,0] should be ALEPH_0.

All the best,
R



"Should" is a good word to use. When would aleph_0 hash marks be completed, exactly? At no finite time before noon are all hash marks completed: more remain to be added in the remaining time. But, AT noon, no hash marks can be added, or that would imply an n such that 1/n=0, which cannot be a finite n. If it cannot be completed before noon, and cannot be completed at noon, how can it be completed by noon?

Happy Trails,

Tony
.

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