On 12/6/2013 3:11 PM, WM wrote: > Am Freitag, 6. Dezember 2013 20:11:32 UTC+1 schrieb Michael F. Stemper: > >> >> (For starters, did >> >> Cantor include the assumption that the list had all reals?) > > No. He said: Ist E1, E2, ..., E?, ... irgendeine einfach unendliche Reihe von Elementen der Mannigfaltigkeit M > Let ... be any infinite sequence of elements ... >
You are correct with regard to the 1874 paper. Ewald has his earlier statements to Dedekind of 1873 phrased differently.
The paper has the proof previously posted by Virgil:
> ********************************************* > A PROOF OF THE UNCOUNTABILITY OF THE REALS > (A variation on Cantor's FIRST proof) > > ASSUMPTIONS: > > (1) the intersection of a strictly nested sequence of closed real > intervals (the endpoints of each interval being interior points of the > previous interval) is not empty. > > (2) A strictly increasing sequence of naturals does not have a natural > as its limit, > > (3a) A strictly increasing but bounded sequence of reals has a real > number as a limit, its least upper bound, different from every member of > the sequence. > > (3b) A strictly decreasing but bounded sequence of reals has a real > number as a limit, its greatest lower bound, different from every member > of the sequence. > > Proof: > > If the set of all reals is countable then we may assume each real can be > and has been paired with a natural so that different reals are paired > with different naturals with none of either left out. > > Assuming this has been done, take the two reals corresponding to the > lowest naturals as endpoints of a real interval. > > It is clear that all the interior points of this real interval must be > paired with naturals larger than those naturals paired with its > endpoints. > > Now take the two reals INTERIOR to the previous interval with the lowest > naturals to be the endpoints of a subinterval of that interval. > > It is clear that the interior points of such a real interval must be > paired with naturals larger than the naturals paired with its endpoints. > > By repeating this process of forming new intervals one generates a > decreasing, but never empty, sequence of closed real intervals each of > which contains only points with higher attached naturals than its > endpoints have. > > The intersection of such a nested sequence of closed intervals is not > empty, but the natural associated with any of its members is necessarily > larger than all of the infinitely many natural numbers associated with > those infinitely many endpoints. > > But there is no natural number larger than infinitely many different > natural numbers. > > This is a contradiction which can only have been caused by our original > assumption that the reals were countable, so it proves they are not > countable. > > QED! > > NOTE: While I have never seen this particular proof in the literature of > countability, it is so obvious that I doubt that it is original with me. >