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fom
Posts:
1,968
Registered:
12/4/12


Re: Matheology § 203
Posted:
Feb 12, 2013 3:21 PM


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.fhaugsburg.de> writes: >>> On 4 Feb., 13:35, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote: >>>> WM <mueck...@rz.fhaugsburg.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 selfreferencing > 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 selfreferencing definitions...
Your observation that the natural numbers are in onetoone 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.



