
§ 424 Actual Infinity: We never get it  but we get it!
Posted:
Jan 28, 2014 3:24 AM


In actual infinity the set N of all natural numbers exists as a union such that no natural number is missing. Considering the union stepwise
{1} {2, 1} {3, 2, 1} ... {..., 3, 2, 1} = N
we see that N does not appear in any enumerated line, i.e., it appears never, but it appears according to the motto: It doesn't matter that we never get it  if only we get it. So the sequence somehow has to reach, create, or complete its limit.
This case can be translated into analysis. If actual infinity applies here, then the sequence
0.000 0.1000... 0.11000... 0.111000... ... 0.111...
reaches, creates, or completes its limit with an actual infinity of digits too.
This means that Cantor's diagonal argument fails in case the antidiagonal d of the sequence is chosen to be d = 0.111... Of course d is not completed in any enumerated line but only in the infinite  alas there it is already welcomed by itself.
Usually set theorists deny that d belongs to the infinite list. Therefore the projection of d on the horizontal axis is never completed (that would require a completed line). But the projection on the vertical axis is completed. And from that part it is concluded in reverse that d is completed. Only by this incoherent argument it is possible for d to differ from every line.
Not necessary to mention that in analysis this limit is not created by digits. We have to use finite definitions for what we never get by digits. The above list does never reach, create, or complete a string of digits without a tail of infinitely many zeros. And in analysis "never" means never.
Regards, WM

