In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> 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.
So do you!
> And you contradict the fact, that in incomplete sets we cannot know > the quality of the complete set.
But outside of WMYTHEOLOGY we can and do know the complets setS , |N and |Q and |R.
Without the ocmplete set |R, analysis is impossible.
> 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?
If there are lines in it that do not exist, then whatever you have, it is not a list.
A list is, by definition, a mapping from some non-empty but not necessarily finite initial segment of naturals to some set. > > > > 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.
Only in WMytheology. Outside WMytheology things are different. > > > > 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.
Then even in WMYTHEOLOGY the set of reals is not "potentially" infinite, since it is provably NOT COUNTABLE, i.e., provably contains more than countably many members. It is equally provably not finite and does exist. > > Every potentially infinite decimal is a countable set of rational > approximations only.
Only in WMytheology.
> 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.
Then only countably many reals have finite definitions, but that in no way makes the others non-existent, at least not outside WMytheology.
> (Actually infinite is tantamount to more than every finite. Aleph is > larger than every n.)
Just so. > > So you accept actual infinity in decimals, but not in lists?
Certainly in lists, at least in lists of members of actually infinite sets like |N and |Q and |R.
> Impossible. If the set of indexes is actually infinite in decimals, > then it is actually infinite in line numbers of lists.
Then the line numbers of such lists exhaust the actually infinite set |N.
Note that every function having domain |N is, by definition, an infinite list. --