In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 5 Feb., 21:55, William Hughes <wpihug...@gmail.com> wrote: > > On Feb 5, 6:02 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > <snip> > > > > >... one cannot even put all natural numbers in > > > correspondence with all natural numbers. > > > > However, one can put every natural number in > > correspondence with every natural number. > > One can have that by belief or decreed by the powers that be. > n <--> n is the finite expression of this belief. But its validity is > dubious because one cannot have possibly in a possible list that > contains every finite initial sequence of the sequence of natural > numbers the sequence s of every finite initial segment of the sequence > of natural numbers.
If one cannot even justify an n to n correspondence for naturals n, then one has thrown out a large baby with a very little bathwater, as one has automatically thrown out induction as well.
Induction requires that one be able to conclude that something holds for all n in |N.
But if one cannot anything for all n in |N, no induction! --