In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
I assume that in the following , the set D represents something like the set of all FISONs of |N. so that the union of D is |N.
> > Ok, we have now established that if you remove > > any finite subset of D, something remains.
Let f:D ->D be any injective function then the union of f(D) is also |N, and one may remove all of D\f(D) without affecting the union of what is left. > > Of course. Even if you remove all finite lines, something remains, > given that D is more than all finite lines.
In WOLKENMUEKENHEIM , one can apparently remove every member of a set and still have a non-empty set. > > > > My claim > > > > If you remove any finite subset of D, what > > remains contains every natural number. >
> > With no doubt. The question is only whether D is actually infinite > (containing more than the union of all finite lines)
That is not an acceptable definition of actually infinite, at last not anywhere outside Wolkenmuekenheim.
OUTside of Wolkenmuekenheim an infinite union of finite sets, which WM has just allowed to be finite, need not be finite.
> > Again the question remains: Is the union of all FISONs of a path more > than all paths or not?
The union of all FISONs of a path in any tree is that path. > > To put it somewhat easier (for readers like Virgil and fom): > (1) Does the sequence of decimals > 0.1 > 0.11 > 0.111 > ... > contain 1/9?
Since those decimals are not SETS, which FISONs are by definition, they cannot be unioned like FISONs can.
The sequence of decimals indicated does converge to 1/9, but convergence in that sense is something quite different from unions of sets, as any competent mathematician should have known. --