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Re: Matheology § 162
Posted:
Nov 27, 2012 6:03 PM
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On Nov 27, 5:17 pm, WM <mueck...@rz.fh-augsburg.de> wrote:
> On 27 Nov., 21:48, William Hughes <wpihug...@gmail.com> wrote:
<snip>
> > > but that does not mean that all
> > > positions are unoccupied:
>
> > But this is not what I am saying.
This is an important point. Clearly "each n in |N has
property P" does not mean "all n in |N have property P"
if we assume that "all n in |N" might not exist.
However from "each n in |N has property P" we can
conclude that no n in |N has property (not P).
> > Compare
>
> > Each natural number >1 is either prime or composite.
> > The set of natural numbers >1 that are neither prime
> > nor composite is empty.
>
> Yes, but here we have a property that is a property of the set like:
> Every natural number is a natural number.
Induction works for any property P. E.g Each natural number
is a natural number can be proved by induction.
1 is a natural number.
If n is a natural number n+1 is a natural number
Each n in |N is a natural number.
One gets the same "contradictions" if one uses potential
infinity rather than actual infinity.
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