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Topic: Matheology § 203
Replies: 4   Last Post: Feb 2, 2013 4:28 PM

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Posts: 18,076
Registered: 1/29/05
Re: Matheology § 203
Posted: Feb 2, 2013 4:52 AM
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On 2 Feb., 02:56, Alan Smaill <> 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, 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.
Compare Matheology § 205 here_
> You only ever have finitely many of them, so you can never know
> what will happen when you look at a new one.

The new one is finite and not more than 1 different from its
predecessor. And there are never more than finitely many. That's
enough to apply the above formalism.

Thank you for that question.

Regards, WM

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