On 28 Nov., 13:48, William Hughes <wpihug...@gmail.com> wrote: > On Nov 28, 2:43 am, WM <mueck...@rz.fh-augsburg.de> wrote: > > > Induction proves also: Every set of natural numbers is finite. > > Why do you overlook this simple proof? > > No, what induction proves is that every set of natural numbers > with a largest number is finite.
And induction proves that every set of natural numbers has a largest number. For every finite n also n + 1 is finite and the set containing both, n and n + 1 ist finite too.
> Induction is agnostic on > the question of whether there can be a set of natural numbers > without largest number.
That's because such a set is a contradiction. Every number has a magnitude, i.e. it is larger or smaller than another number. A set of naturals without largest number is like a number without numerical value or a word that can't be spelled.
> The interesting thing is that > you get the same "contradictions" whether or not you allow a > set of natural numbers without largest number.
You are wrong. But that is not under discussion. Under discussion is the fact that set theory and analysis deliver different results. Or do you continue to claim that analysis cannot determine that the limit of my sequence has more than zero digits?