On 15 Jun., 21:03, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > WM says... > > >On 15 Jun., 18:46, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > >> I'm not sure what you are saying. The fact is, we can prove > >> that for every real r_n on the list, d is not equal to r_n. > > >Of course. Every real r_n belongs to a finite initial segment of the > >list. > >That does not yield any result about the whole list > > On the contrary, the definition of "d is on the list" > is that "there exists a natural number n such that r_n = d". > We proved "forall n, r_n is not equal to d". So that > means "there does not exist a natural number n such that r_n = d", > so that means "d is not on the list". > > We have thus proved something about the whole list. > > >> That means that d is not on the list. There is no extrapolation > >> involved. > > >Look here: We can prove for any finite segment > >{2, 4, 6, ..., 2n} > >of the ordered set of all positive even numbers that its cardinal > >number is surpassed by some elements of the set. > > >Nevertheless this appears not be a proof that the cardinal number of > >the whole set is less than some elements of the set. > > So there we have an example of an illegitimate extrapolation. > If you prove Phi(n) for an arbitrary natural number n, then you are allowed > to conclude: > > forall natural numbers n, Phi(n). > > So you can conclude: > > forall natural numbers n > 0, the set of all even numbers less than or > equal to 2n has a cardinality less than 2n. > > That's true. That's a legitimate proof. On the other hand, it is not > legitimate to conclude: > > The set of all even numbers has a cardinality that is less than > some even number. > > That's an unwarranted extrapolation. > > So there are legitimate proofs, and there are bogus proofs.
And Cantor's proof shows something for every natural number: Every line numerated with a natural number from 1 to n does not contain the diagonal. But to conclude that the whole set, i.e., the infinite list, does not conclude the diagonal is a bogus conclusion. Not a legitimate proof.
