In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> 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.
In my world proving something "for all naturals numbers" proves it for all elements of the set of natural numbers. > > > 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.
Since everywhere but inside Wolkenmuekenheim there is already a perfectly adequate defnition of a set being finite, your attempt revise that definition is irrelevant outside of Wolkenmuekenheim.
> Nevertheless there is no actually finite set of natural numbers.
Actually, every FISON is an actually finite set of natural numbers, it is just that none of those FISONs is a set of ALL natural numbers the way their union is. --