In article <firstname.lastname@example.org>, WM <email@example.com> wrote:
> 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.
Yes it is!!!
> The reason is that omega is a limit > without axiom of infinity
By what definition of "limit"? > > > > 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.
Not outside of Wolkenmuekenheim.
> > > > 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.
In ZF, any union of the sets which are members of a set is "defined" by the axiom of union, and in ZF there is no other form of union at all.
So that what WM is saying about unions is false in ZF. > > > > > 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.