On 27 Okt., 20:06, William Hughes <wpihug...@gmail.com> wrote: > On Oct 27, 10:27 am, WM <mueck...@rz.fh-augsburg.de> wrote: > > > On 27 Okt., 12:04, William Hughes <wpihug...@gmail.com> wrote: > > > > <snip yet another version of your latest > > > "proof" that "actual infinity" is inconsistent> > > > > Whether b_oo exists or not we have C=N=L. > > > This is easily proven wrong with decimal sequences: > > 0.111... has more digits than all its finite approximations. > > Nope.
? You want to say here that aleph_0 is not larger than all natural numbers? Or do you want to say that 1/9 as decimal fractions has less than aleph_0 digits?
> There is one finite approximation for each element of N. > Each finite approximation adds one digit. > 0.111... has one digit for each element of N
Please confirm: 0.111... has not more digits than all finite strings of 1's.
> > > Finite means: there are a number of 1's and then an infinite number of > > 0's following. Obviously the union of all finite sequences of 1's > > cannot yield a number that is larger than all finite sequences of 1's. > > This does not mean anything as written. Please be precise. > E.g. if you mean > > forall n in N: 0.111... is a larger real than 0.11...1 (n terms) > > write this.
No, I do not mean this. I mean omega is an ordinal number that satisfies: Forall n in |N : n < omega. > > > (If it would, then b_oo could be found in the union of all lines, > > which you agreed is not the case.) > > Piffle. b_oo is the union of all lines, and I have never claimed > anything else
So you mean that for the number of digits of b_oo, let's call it aleph_0, we have
Exists aleph_0 Forall n in |N : n < aleph_0 is incorrect? But correct is only: Forall n in |N Exists a(n) : n < a(n)?