```Date: Feb 4, 2013 7:35 AM
Author: Alan Smaill
Subject: Re: Matheology § 203

WM <mueckenh@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?  >> 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.This is justification by fiat, the last refuge ofthe Matheologists.  When in doubt, say that there is no doubt.So, WM take this as an axiom of WMathematics.(1 + 1 = 2 is purely computational;  induction overformulas of abritrary complexity, say with several quantifiersis a whole different affair)> Compare Matheology § 205 here_> http://www.hs-augsburg.de/~mueckenh/KB/Matheology.pdf>>>> 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.But the conclusion tells us that there is a property that holds for*every* natural number (not all) -- some of which by your accountwill never come into existence at all (otherwise we wouldthen have all of them)."for every natural number n, if n is odd then n^n is odd"You will run out of ink to write down n^n pretty quickly.When n is available, but not n^n, you are lapsing into theology.Nelson's attitude on exponentiation is different.But no doubt you disagree with him also.> Regards, WM-- Alan Smaill
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