On Saturday, July 13, 2013 7:40:24 AM UTC-7, Julio Di Egidio wrote: > "Zeit Geist" <email@example.com> wrote in message > > news:firstname.lastname@example.org... > > > On Friday, July 12, 2013 1:41:31 PM UTC-7, muec...@rz.fh-augsburg.de > > > wrote: > > >> On Friday, 12 July 2013 19:13:19 UTC+2, Zeit Geist wrote: > > >> > > >> > It is rather silly to expect the process that creates each of the > > >> > Naturals would produce the set of all Naturals, as that set is, > > >> > itself, not a Natural. > > >> > > >> Each natural belongs to a finite initial segment. None of them > > >> requires a number that is larger than every natural number. In > > >> fact the contrary. If you do not talk about the set, then there is > > >> no reason to talk about alephs. > > > > > > Yes, but for every Natural there is a larger natural, hence the number > > > of Naturals is larger than any Natural. > > > > Since the number of natural numbers is not itself a natural number, that is > > a non-sequitur, despite standardly the conclusive statement is correct: > > indeed, a fallacy of relevance. Plus, the standard here is in question, so > > one should rather qualify statements as well as objections (not that WM ever > > does it, of course). >
The are numbers that are not Natural Numbers. The number of Naturals Numbers is a number, and it greater than any finite number, that is to say, It is greater than any Natural Number.
Here, number means Cardinality, of course.
In most Mathematical circles the standard is ZF(C). Yes, standard Set Theory is being questioned here. And most who question it here have not come up with a good reason to reject. Nor have they come up with a suitable replacement.