On Tuesday, 11 June 2013 20:23:17 UTC+2, Julio Di Egidio wrote: >> After each one has been > attached to a natural, we have the set of naturals that obviously can > be re-ordered in any desired way.
> Obviously and crucially not in any finite number of steps (you will get the rationals well-ordered by magnitude from, say, the rationals lexicographically ordered). Anyway, I'm already out of my depths here, so I'll leave this to more competent mathematicians.
It is obvious: If we show in set theory a proposition P(n) for the first n elements of a well-ordered set (where n is an arbitrarily large natural number), then we do show it for all elements of the set.
If we enumerate the rationals, we do it up to n (since there is no infinite natural number). If we apply the diagonal argument, we do it up to n (since there is no infinite natural number). If we well-order the rational numbers by magnitude, we do it up to n.
There is absolutely no difference.
> (The power set of |N should even include the singleton of the last natural > as an element.) There is no such thing as the last natural number, and not even a next to last, and not even a next to next to last, and so on.
I know. But there are more than any natural number of numbers. So if we take the numbers 1 to n for any natural number, then we have less than aleph_0 naturals. What remains to take all?
This is but *one* simple aspect which shows that the jerks of matheology should really be imprisonde in a mad-house.
It's a bit of a pain to have to reformat your posts and quotes every time: it would help a lot if you could at least split your paragraphs in short lines using carriage returns.
That's due to the new Google which is an outspoken shit. Unfortunately it seems that I cannot return to the old version. The developer of that mess be cursed.