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? Countability needs to be effective : That is , you need to actually provide a way (finite description) to count your stuff . I can make a function from N to the prime numbers , thus the prime numbers are countable .
Let's say you construct an enumeration from N to the set S of pairs (d, n) of a date ,and a lottery number . You proved the set of all possible dates and lottery numbers is countable . Now , within this set , we have W , the set of "winning dates and lottery numbers") . If you manage to count W , give me a call .
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 .