Re: An infinite debate


"Tony Orlow" <tony@xxxxxxxxxxxxx> wrote in message
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RLG wrote:
"Tony Orlow" <tony@xxxxxxxxxxxxx> wrote in message
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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?

You could add a hash mark at noon, say at 0, in the interval [-1,0]. But
that would not change anything since 1+ALEPH_0 equals ALEPH_0.

R


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    ... Tony Orlow wrote: ... 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. ... But what if by `n' we mean the number of hash marks that could be made in a unit interval by noon? ... At noon, no element can be added to complete the sequence, yet, at any time before noon, the sequence cannot have been completed, since there are more to be added during that finite interval. ...
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