Re: An infinite debate

David Marcus wrote:
Tony Orlow wrote:
William Hughes wrote:
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.

T_c is a set of *times*. So, T_c is a set of real numbers between -1 and 0. Are you denying that a set of real numbers between -1 and 0 has a supremum?


Uh, no, but I am denying that there is a supremum or LUB of N, which is mapped to T_c.

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.

What does it contradict?

The fact that, if t1<t2 and at t1 the set is not complete and at t2 it is, then there exists a t3 such that t1<t3<=t2 when the set became complete.


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.

Exactly. You've got it!


Except that in order for the sequence to go from incomplete to complete, elements must be added to it. Do you disagree with that?
.

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