On 2/12/2013 11:21 AM, WM wrote: > 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.
Speaking of self-referencing definitions...
Your observation that the natural numbers are in one-to-one correspondence with the initial segments is invoking the successor as a choice function
It is one of the truly elegant observations in your analysis. Sadly, you cannot see it because of your religion.
It also suggests that all arithmetic is impredicative.