In article <firstname.lastname@example.org> WM <email@example.com> writes: > 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.
I have never seen there a proper definition of omega.
> > There are no concepts of mathematics without definitions. > > So? What is a set?
Something that satisfies the axioms of ZF for instance.
> > > 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.
Yes, so what? You are arguing against current set theory, in the time of Cantor it was still being developed.
> > 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.
If a function exists as a limit, then it has gotten from the finite to the infinite one by one step. During this process there is no chance of any divergence between the function and the integral.
Now, what is wrong with that reasoning?
Stronger: lim(n -> oo) 1/n = 0 1/n > 0
If a number exists as a limit, then it has gotten from the finite to the infinite one by one step. During this process there is no chance of any divergence between the element and the inequality.
What is wrong with that reasoning?
You are assuming that taking a limit is a final step in a sequence of steps. In the definition I gave for the limit of a sequence of sets there is no final step. -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/