On 11 Dez., 03:40, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: > In article <5333fb9a-1670-4fcc-85d3-25e75fb5b...@f16g2000yqm.googlegroups.com> WM <mueck...@rz.fh-augsburg.de> 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. > I knew you did not read it.
Look at p. 93. There the natural numbers are constructed. On p. 86 all ordinals till eps^eps^eps^... are given. On p. 90 you see the axiom of infinity 7.
> > > There are no concepts of mathematics without definitions. > > > > So? What is a set? > > Something that satisfies the axioms of ZF for instance.
Is that a definition?
But in case you shouldn't have been able to find a definition of actual infinity, here is more than that: omega + 1.
> > 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
I do not understand. Do you see a gap here? > > 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.
I did not say that there is a final step. I say that there is no chance for a difference of lim card(S_n) and card(lim(S_n)) where lim means n --> oo.
If, in your example you would claim that lim(1/n) = 0 and and 1/omega = 10, I would not accept such behaviour as mathematics (like your funny Sum n = 0 or Euler's -1/12).