On 1 Feb., 16:38, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote: > William Hughes <wpihug...@gmail.com> writes: > > Let P(n) be > > 0.111... is not the nth line > > of > > > 0.1000... > > 0.11000... > > 0.111000... > > ... > > > Clearly for every natural number n > > P(n) is true. > > > This means there is no natural > > number m for which P(m) is true. > > > It is not simply that we cannot find m, > > we know that m does not exist. > > Futhermore WM accepts, for example, that > for every natural number n, 2 * n is even. > > Doesn't he?
Of course. > > But when asked how it is possible to know such a thing, > he falls strangely silent.
For that theorem you need not know (actually) all natural numbers. Induction is sufficient that holdes for (potentially) all natural numbers, i.e., up to every natural number. There is no impredicative definition involved.