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Re: Matheology § 203
Posted:
Feb 12, 2013 12:21 PM
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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.
Regards, WM
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