In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> On 10 Dez., 16:29, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > > > Without the axiom of infinity omega would not be immediately existing. > > So apparently there is a definition of omega without the axiom of infinity. > > Can you state that definition? > > Look into Cantor's papers. Look into my book.
Cantors papers might be worth it, but no serous mathematician need bother with any book written by WM. Wm has sufficiently often proved his mathematical incompetence here to obviate any need to delve further into it. > > > > There are no concepts of mathematics without definitions. > > So? What is a set?
Does the absence of a definition of "set" imply the absence of all definitions from mathematics? If not then WM's question is, as usual, totally irrelevant.
> > > > > > An infinite union *is* not at all. But if it were, it was a limit. > > > > It *is* according to one of the axioms of ZF, and as such it is not a limit. > > It *was* according to Cantor, without any axioms.
But Russell's paradox showed the need for something like axioms. > > > Where? Why do you think taking a limit and taking cardinality should > > commute? Should also the limit of te sequence of integral of functions > > be equal to the integral of the limit of a sequence of fuctions? > > If an infinite set exists as a limit, then it has gotten from the > finite to the infinite one by one element. During this process there > is no chance for any divergence between this set-function and its > cardinality.
WM offers no proof that infinite sets can exist ONLY as limits, nor how they must be formulated if they are limits, so, as usual, the rest of his speculations are irrelevant.