In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> > > Therefore Cantor "proves" the uncountability of a countable set. > > > > No. Cantor's proof is not about a "set" of Reals; it is about a "list" of > > Reals. > > But from this list he concludes that the set of reals that can appear > as a diagonal, hence can be specified somehow, is uncountable.
> This > conclusion is wrong
YOUR conclusion is not what Cantor argues. What Cantor argues, successfully, is that any 'list' of reals omits some reals. A valid extension of that argument says that any such list omits as many reals as it includes.
Since by definition, "listability" = "countability", Cantor's proof of unlistability proves uncountability. > > > > You seem to not understand that something can be a set but cannot be > > enumerated as a list, which is pretty much the whole point of Cantor's > > proof. > > > I understand that aleph_0 is considered to be a number. Then we can > conclude that even sets that cannot be enumerated, can have cardinal > number aleph_0. Because they are a subset of a set that can be > enumerated.
Relevancy tothe issue of Cantor's proof?
> > > Another proof shows: The reals that can be constructed, are countable. > > > So there is a contradiction. What is the reason: The assumption of > > > finished infinity. > > > > No, the pretty obvious (and correct) conclusion is that you cannot form a > > list of the constructable Reals. > > > > Note, again, that you are implicitly treating a list of Reals as the same > > thing as a set of Reals; Cantor's proof is about Lists, not Sets. > > Cantor concludes about sets. That is wrong.
If infinite countability is equivalent to "listable" but not finite, and Cantor proved the reals not listable, how is it that he has not proved them not countable?
> > > > Well, mathematicians don't confuse lists of Reals with sets of Reals. > > Why then do they claim that the set of reals is uncountable and > conclude that 2^aleph_0 is larger than aleph_0?