Virgil
Posts:
8,833
Registered:
1/6/11


Re: Matheology � 201
Posted:
Jan 28, 2013 2:16 PM


In article <d89b0fecb2464f8bbbb04293d5ca83ed@k4g2000yqn.googlegroups.com>, WM <mueckenh@rz.fhaugsburg.de> wrote:
> 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.
He appears to understand a great deal better than WM does.
> For every natural n, P(n) is true does > not imply anything about the existence of P(m).
"i. for every natural number n, P(n) is true" certainly implies the existence of a lot of P(m)'s as well as their truth.
> You have not yet > grasped that there is no "for all". There is outside of Wolkenmuekenheim. > > 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".
If there is "no natural beyond all", then we already have "all naturals". At least outside Wolkenmuekenheim
> All > we can do is state P(n) for every n. Nevertheless it is not > necessarily true for all n.
So WM claims he can find an instance in which For every n in N P(n) and for some n in N not P(n)
Note that outside of WN's Wolkenmuekenheim [NOT for all in N P(n)] if and only if [Exists n in N such that not P(n)] 

