On 12 Feb., 17:59, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote: > WM <mueck...@rz.fh-augsburg.de> writes: > > On 4 Feb., 13:35, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote: > >> WM <mueck...@rz.fh-augsburg.de> writes: > >> > 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, on what grounds do you suppose that the notion > >> of natural number is predicative? > > > The notion of every finite initial segment is predicative because we > > need nothing but a number of 1's, that are counted by a number already > > defined, and add another 1. > > It's in the justification of the claim that induction yields a conclusion > that holds for *any* natural number where the impredicativity lies.
Impredicativity is not a matter of quantity but of self-referencing definition. Further you seem to mix up every and all. I don't see any necessity to consider *all* natural numbers. I maintain, without running in danger to be contradicted: For every natural number that can be reached by induction, induction holds.