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Virgil
Posts:
9,012
Registered:
1/6/11


Re: Matheology � 203
Posted:
Feb 12, 2013 5:07 PM


In article <9d3a275d15c0440895a5b8b4d11d5a36@e11g2000vbv.googlegroups.com>, WM <mueckenh@rz.fhaugsburg.de> 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 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, 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.
Not nearly so badly as WM does.
> I don't see any necessity to consider *all* natural numbers.
Without considering all of them, one cannot distinguish finiteness from nonfiniteness.
A set is finite if every ordering of it has a last member. A set is infinite if some ordering of it has no last member. In either case it is legitimate to speak of 'every member' or of 'all members' of the set. 



