In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> 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 number¹ > > 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.
That inductive argument appears to be based on the very same flaws that WM objects to in allowing actual infiniteness. > > > > You only ever have finitely many of them, so you can never know > > what will happen when you look at a new one. --