On 30 Jan., 10:13, Virgil <vir...@ligriv.com> wrote: > In article > <b79952f1-a65c-4b62-9cb4-5a358b78b...@4g2000yqv.googlegroups.com>, > > WM <mueck...@rz.fh-augsburg.de> wrote: > > You can prove something for all natural numbers, but not for a larger > > set. > > You can prove that the set of naturals can be injected into a proper > subset of itself. n --> n+1 is such an injection.
In fact this property only shows potential infinity. You prove something for every n but not for all elements of the set.
> Any set of objects > with this property (of being injectable to a proper subset of itself) is > by definition actually infinite.
So what? Similarly we can define: Every set of more than ten natural numbers and sum less than 5 is by definition actually finite. Nevertheless there is no actually finite set of natural numbers.