The last element is not fixed but is a function of a variable about we don't yet know much. It may depend on time, ressources and the distribution of them and on much more. But it is not necessary to introduce reality and psychology into mathematics, as long as we accept potential infinity sinply as inexhaustibility. > > The set, M, of naturals contained in the elements of T is a > potentially > infinite set.
The number of naturals in M(t) is the number of numbers in the temporarily last FISON F(t). It is equal to the number of FISONs in M(t). > > For every natural n, we can find a FISON, F(n), so that n is in F(n).
And for every function of that natural like 10^10^n. > > Every natural is in M > > M is the potentially infinite set N.
M(t) is a finite set with a last element m. Therefore the FISON F_m which contains m is the only necessary FISON - until w takes the place of m, then F_m is no longer necessary but F_w is the necessary FISON. > > Nope, introducing a Wolkenmuekenheim special, a set with last element > but without fixed last element, does not change things.
Independent of my above explanation we can prove that *by the definition of FISON* there are never two or more FISONs required to contain any set of naturals.
The assumption that infinitely many would do is absolutely unjustified. You could also say that infinitely many two-digit-numbers would lead to a 10-digit-number or that parallels would touch in "the infinite". There is a big difference between "never" and "in the infinite".
