
Re: Matheology § 201
Posted:
Jan 28, 2013 3:45 AM


On 28 Jan., 08:44, William Hughes <wpihug...@gmail.com> wrote: > On Jan 27, 11:23 pm, William Hughes <wpihug...@gmail.com> wrote: > > > > > > > On Jan 27, 11:16 pm, WM <mueck...@rz.fhaugsburg.de> wrote: > > > > On 27 Jan., 23:02, William Hughes <wpihug...@gmail.com> wrote: > > > > > On Jan 27, 10:39 pm, WM <mueck...@rz.fhaugsburg.de> wrote: > > > > > > On 27 Jan., 21:40, William Hughes <wpihug...@gmail.com> wrote: > > > > > > > On Jan 27, 6:46 pm, WM <mueck...@rz.fhaugsburg.de> wrote: > > > > > > > > On 27 Jan., 18:32, William Hughes <wpihug...@gmail.com> wrote: > > > > > > > > > On Jan 27, 6:05 pm, WM <mueck...@rz.fhaugsburg.de> wrote: > > > > > > > > > <snip> > > > > > > > > > >..the diagonal > > > > > > > > > cannot differ from all lines > > > > > > > > > (it differs from every line, though). > > > > > > > > > The fact that the diagonal differs from every line is > > > > > > > > enough to show (induction) that the diagonal is not > > > > > > > > equal to any line in the list. > > > > > > > > No. > > > > > > > Let the antidiagonal be d and the nth line be l(n) > > > > > > > We know that for each n in N, d is not equal to l(n) > > > > > > > You have agreed that this implies > > > > > > > There is no m in N such that d equals l(m) > > > > > > No. > > > > > You contradict yourself. You have agreed > > > > that if P(n) is true for every n then > > > > the is no n such that P(n) is false. > > > > Don't turn the words in my mouth. > > > You have apparently forgotten the > > thread in which you agreed to this.
I did not agree to that, > > A simple proof. > > The statements > > i. for every natural number n, P(n) is true > ii. there exists a natural number m such that P(m) > is false > > cannot both be true at the same time. > If you prove that i. is true then it follows > that ii. is false.
You have not yet understood. For every natural n, P(n) is true does not imply anything about the existence of P(m). You have not yet grasped that there is no "for all".
Of course for every n it is true that there are infinitely many naturals beyond. Of course there is no m such that there is no natural beyond. But if you assume that all natural exist, then there is no natural beyond all.
You must unederstand that, therefore, there is no "all naturals". All we can do is state P(n) for every n. Nevertheless it is not necessarily true for all n.
Regards, WM

