In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 8 Dez., 19:36, A <anonymous.rubbert...@yahoo.com> wrote: > > > > > A function f is said to be continuous at a point x in its domain if > > the limit of f(a), as a approaches x, is equal to f(x); in others > > words, the limit of the values of f is equal to the value of f at the > > limit, speaking loosely. Of course, not every function is continuous > > at every point in its domain, and some functions are not even > > continuous at any point in their domains at all. > > > > The situation for sets and cardinality is no more mysterious than > > that. The cardinality of a limit of subsets of the integers is not > > guaranteed to be the limit of the cardinalities of those subsets. You > > don't expect an arbitrary function to always be continuous, so perhaps > > it's unreasonable to expect the cardinality "function," defined on > > subsets of the integers, to be continuous.- > > That depends on the circumstances. If infinite sets exist, then they > have a cardinality.
No one says otherwise, but that in no way implies that the cardinality of the limit of a sequence of sets as defined by DIk need equal the limit, if it even exists, of the cardinalities of those sets.
Such a claim requires proof, which WM has been unable to provide, and such a claim cannot stand in view of the counterexample which Dik provided, so that WM is doubly wrong, in (1) not having a proof of his claim and (2) having found no flaw in Dik's counterproof.
> Then the limit cardinality is the cardinality of the limit set.
Often claimed by WM but not proved by him nor has WM, or anyone else, refuted Dik's counterexample. > > If they do not exist but are merely an arbitrary, perhaps inconsistent > delusion, then everything is possible.
In Wolkenmuekenheim, WM may declare to be possible whatever he likes and declare to be impossible whatever he likes, but outside of his private world, he is not master over what is and is not possible, however often he may claim to be.
I argue, based on Cantor's > claim, that infinite sets exist (in order to show that they do not).
At which argument, as with so many others, WM fails miserably.