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Virgil
Posts:
4,482
Registered:
1/6/11
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Re: Matheology � 203
Posted:
Feb 2, 2013 2:24 PM
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In article
<37a55e15-deab-4cc3-9497-1915e997705c@k8g2000yqb.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> 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.
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