In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 7 Apr., 20:32, Dan <dan.ms.ch...@gmail.com> wrote: > > On Apr 7, 9:20 pm, WM <mueck...@rz.fh-augsburg.de> wrote: > > > > > On 7 Apr., 20:00, Dan <dan.ms.ch...@gmail.com> wrote: > > > > > > So , we have a set that you can't count (all meaningful finite > > > > sentences of words , it's simply too complex for you to count) , stuck > > > > inside a set you can count (all random words/sequences of > > > > letters) . > > > > > You need not count a set in order to prove its countability. A subset > > > of a countable set is countable in set theory and wherever > > > countability appears to be a meaningful notion. You cannot save > > > uncountability in set theory after violating this theorem. > > > > > Regards, WM > > > > How can you say a set is countable if you can't "actually" count it? > > "Countably infinity" is the least infinity. A subset of it cannot have > larger cardinality if cardinality should have any meaning. IIRC > otherwise already Schroeder-Bernstein would fail. > > > Countability needs to be effective : > > No. I cannot count the rabbits on earth, nevertheless I know they are > countable. Every meaningful application of set theory needs the > theorem that a subset cannot have larger cardinailty than its > superset. > > But, of course, this yields a contradiction.
Only in Wolkenmuekenheim! > > > > The difference between countable and uncountable is as obvious as the > > difference between playing ALL POSSIBLE lottery numbers and playing > > only the WINNING ones .
Outside of Wolkenmuekenheim, a set is countable if and only if there is surjection from the set of all natural numbers, |N, to the set in question. Thus both |N and each of its subsets is countabla, though the set of all its subsets is not.
> > In set theory we have the theorem that a subset of a countable set has > cardinality aleph_0 or is finite. If you want to introduce "effective > countability" in order to save set theory, you destroy it.
Just as introducing WMytheology to mathematics destroys almost oll of its mathematical validity. --