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. > > The difference between countable and uncountable is as obvious as the > difference between playing ALL POSSIBLE lottery numbers and playing > only the WINNING ones . You can make yourself a fantasy that you only > play THE WINNING NUMBERS (after all, their just a subset of ALL > POSSIBLE NUMBERS) , but that don't make it so .
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.