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Virgil
Posts:
4,482
Registered:
1/6/11
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Re: Matheology � 203
Posted:
Feb 12, 2013 5:07 PM
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In article
<9d3a275d-15c0-4408-95a5-b8b4d11d5a36@e11g2000vbv.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:
> 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 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 self-referencing
> 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
non-finiteness.
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.
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