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.