On 2 Feb., 02:56, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
> "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."
I do not agree with these authors on this point. > > So, how on earth do you know that induction is a correct > principle over the natural numbers?
If a theorem is valid for the number k, and if from its validity for n + k the validity for n + k + 1 can be concluded with no doubt, then n can be replaced by n + 1, and the validity for n + k + 2 is proven too. This is the foundation of mathematics. To prove anything about this principle is as useless as the proof that 1 + 1 = 2. Compare Matheology § 205 here_ http://www.hs-augsburg.de/~mueckenh/KB/Matheology.pdf > > You only ever have finitely many of them, so you can never know > what will happen when you look at a new one.
The new one is finite and not more than 1 different from its predecessor. And there are never more than finitely many. That's enough to apply the above formalism.