In article <email@example.com> WM <firstname.lastname@example.org> writes: > On 8 Dez., 15:22, "Dik T. Winter" <Dik.Win...@cwi.nl> wrote: ... > > > I said you can take it from the shelf. It is not defined as a limit > > > (if you like so) although amazingly omega is called a limit ordinal. > > > Yes, it is called a limit ordinal because by definition each ordinal that > > has no predecessor is called a limit ordinal (that is the definition of > > the term "limit ordinal"). It has in itself nothing to do with limits. > > No, that is not the reason. The reason is that omega is a limit > without axiom of infinity, and omega is older than that axiom.
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?
> > > N is a concept of mathematics. That's enough. > > > > Yes, and it is a concept of mathematics because it is defined within > > mathematics, and it is not defined as a limit. > > It is a concept of mathematics without any being defined.
There are no concepts of mathematics without definitions.
> > > The infinite union is a limit. > > > > I do not think you have looked at the definition of an infinite union, if > > you had done so you would find that (in your words) such a union is found > > on the shelf and does not involve limits. Try to start doing mathematics > > and rid yourself of the idea that an infinite union is a limit. > > 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.
> > > Why did you argue that limits of > > > cardinality and sets are different, if there are no limits at all? > > > > I have explicitly defined the limit of a sequence of sets. With that > > definition (and the common definition of limits of sequences of natural > > numbers) I found that the cardinality of the limit is not necessarily > > equal to the limit of the cardinalities. > > That means that you are wrong.
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? -- dik t. winter, cwi, science park 123, 1098 xg amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/