In article <ee9de39b-0511-4088-a62a-2febd5424b43@c10g2000yqi.googlegroups.com>, WM <mueckenh@rz.fh-augsburg.de> wrote:
> 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.
By WM's argument, proofs by induction are all invalid. So that I, for one, will choose to reject WM's argument rather than rejecting inductive proofs. > > > > 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.
By WM's argument, proofs by induction are all invalid. So that I, for one, will choose to reject WM's argument rather than rejecting inductive proofs.
