> 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.
Nothing is "of course" when the Prophet speaks.
>> 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.
"The logicist reduction of the concept of natural number met a difficulty on this point, since the definition of ?natural number? already given in the work of Frege and Dedekind is impredicative. More recently, it has been argued by Michael Dummett, the author, and Edward Nelson that more informal explanations of the concept of natural number are impredicative as well. That has the consequence that impredicativity is more pervasive in mathematics, and appears at lower levels, than the earlier debates about the issue generally presupposed."
So, how on earth do you know that induction is a correct principle over the natural numbers?
You only ever have finitely many of them, so you can never know what will happen when you look at a new one.