On 29 Jan., 21:28, William Hughes <wpihug...@gmail.com> wrote:
> > Here we have again the ambivalence required for set theory. No, your > > statement is incorrect if "infinite" is used in the sense of completed > > or actual, > > It is not. No concept of "completed" is needed or used.
You contradict Cantor. And you contradict the fact, that in incomplete sets we cannot know the quality of the complete set. But in the first place, look at the end of this posting. > > It does, however, imply that d in not one > of the lines of the list L
For that sake you must check all lines. Can you check what is not existing? > > This is turn implies. > > There is no list of binary sequences, L, with > the property that give any binary sequence, s, > s is one of the lines of L.
That is wrong in the list of all binary sequences that can be defined or identified. > > So we can divide collections into two groups. > Those, like the collection of all rational > numbers, that can be listed, and those > like the real number that cannot.
Here you apply actual infinity. Every potentially infinite set consists of countably many finite elements.
Every potentially infinite decimal is a countable set of rational approximations only. In order to obtain the true decimal of an irrational number you will need a finite definition like "pi" (unfortunately there are only countably many finite definitions available) or a sequence of digits that is longer than every finite sequence. But that means, the length of the sequence is actually infinite. (Actually infinite is tantamount to more than every finite. Aleph is larger than every n.)
So you accept actual infinity in decimals, but not in lists? Impossible. If the set of indexes is actually infinite in decimals, then it is actually infinite in line numbers of lists. Then we get C4 is wrong.