Re: An infinite debate

William Hughes wrote:
Tony Orlow wrote:
William Hughes wrote:
Tony Orlow wrote:
RLG wrote:
"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


I'm not sure how that answers the conundrum. It seems you admit it
really doesn't. 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.
So, at what point in time is the set completed?
There is no point in time at which the set is completed.

- William Hughes

Then how can you say the event of completion occurs? It does not happen
before noon, nor at noon, so it does not happen by noon. Is that wrong? Why?

Yes that is wrong.

Let T_c be the set of all times at which an element is added to the
sequence. T_c is bounded above and below, so T_c has
both an infinmum and a supremum. The infimum is an element
of T_c, so T_c has a minimum. The supremum is not an
element of T_c so T_c does not have a maximum (This
can only occur if T_c has unboundedly many elements).

Since T_c does not have a maximum,
there is no time at which the event of completion occurs.

Let the supremum of T_c be t_f.

If that is the LUB of T_c, then there really is no such thing, the way I see it. I know you claim omega to be the smallest infinite ordinal, and some sort of a LUB on N, but I rather see that as antihtetical to the notion that adding any nonzero quantity x, positive or negative, to any quantity y, yields a sum z<>y. As a limit ordinal, omega-1=omega, violating this principle. If the basics of addition are upheld, then the conclusion that there is no smallest infinity, or LUB on the naturals, is the only conclusion.


At any time s<t_f the sequence is not completed.
At any time t>= t_f the sequence is completed.

So there is a time, t_f, such that before t_f
the sequence is not complete and by t_f the sequence is
complete.

That implies that the sequence is completed at t_f, except that no elements are added at t_f. That's a contradiction.


In the present case t_f is noon.
So before noon the sequence is not complete, and by noon the
sequence is complete.

Which means it's completed at noon, a moment when no elements are added.


- William Hughes

.

Relevant Pages

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  • Re: An infinite debate
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