On 15 Jun., 16:32, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > WM says... > > > > >On 15 Jun., 12:39, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > > >> That's *all* that matters, for Cantor's theorem. The claim > >> is that for every list of reals, there is another real > >> that does not appear on the list. > > >The claim is only proved for every finite subset of the list. > > The proof does not make use of any property of infinite lists. > The proof establishes: (If r_n is the list of reals, and > d is the antidiagonal) > > forall n, d is not equal to r_n
As every n is finite, it belongs to a finite initial segment of the infinite list. > > There is no "extrapolation" involved. The way that you prove > a fact about all n is this: > > Prove it about an unspecified n.
Specified or not. n is finite anyhow and belongs to a finite initial segment of the list. Only for that always finite segment the proof is correct.
> Use universal generalization. > > There is no extrapolation involved.
Similarly we see that Hercules' list contains no line that is the last one. Hence there is no digit of the decimal expansion of pi that is not contained together with all its predecessors in one single line. If none is not contained, then all are contained, arn't they?