In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 9 Feb., 22:09, Virgil <vir...@ligriv.com> wrote: > > > Does the finiteness of the members of that set establish the finiteness > > of the set itself? > > The finiteness of the natural numbers in combination with the constant > difference of 1 establishes that the natural numbers are in bijection > with the initial segments. This together with principle of induction > implies that the sequence 1, 2, 3, ... has no upper threshold
There are all sorts of ordered sets in mathematics that have no last member, but that does not mean that they cannot be sets. But your "potentially infinite sets", being ambiguous as to membership, are not sets at all.
> > > > What term or terms does WM want to use for > > > > "more than any finite number finite levels"? > > Potentially infinite.
But a set in standard mathematics cannot be thus ambiguous, if its membership is indeterminant, as you would have it, it is not a set of any sort, at least not in standard matheamtics.
Thus the set of all natural numbers is properly a set, but a set that is only potentially all natural numbers is not an actual set until it actually acheives that potentiality. > > > > > The term is infinity, the limit is the same (improper limit) as of > > > the supersequence 1, 2, 3, ... of 2, 4, 8, ..., denoted by oo. > > > > Actually, proper grammar, at least in English, requires that the term be > > "infinite" not "infinity" > > You are not a native speaker? The noun is infinity.
Your phrase "Potentially infinite" acts as an adjective, not a noun in standard English, so don't try to correct your betters. --