On 2/5/2013 5:49 PM, Virgil wrote: > 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! >
Even Wittgenstein's finitism admitted correspondence by finitely specified rules (as opposed to the Dirichlet abstraction of any collection of ordered pairs satisfying the usual constraint of well-definition).