On 4/7/2013 5:09 PM, Virgil wrote: > In article > <firstname.lastname@example.org>, > WM <email@example.com> 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. >
That would require acknowledgement that limits may converge outside of the domain from which their constituents are taken, acknowledgement that an "arithmetic of limits" refers to a system of relations whose relata are outside the domain from which their constituents are taken, and an acknowledgement that the relation between the grounding arithmetical system and the derived arithmetical system of limits is representable as a part/whole relation which has different cardinalities on the basis of Cantor's diagonal argument.
So, it must be time to start "willing" concatenations of monotone inclusive crayon marks again.