On 16 Jun., 19:45, MoeBlee <jazzm...@hotmail.com> wrote:
> But, every countable set does have a bijection with N or has a > bijection with some member of w (the set of natural numbers).
That would be true if countability and aleph_0 were not self- contradicting concepts.
But there is a set that is less than uncountable but has no bijection with N or a definasble subset of N. This set is the set of all finite definitions (in binary representation).
0 1 00 01 ...
where every line may be enumerated by an element of the countable set omega^omega^omega (and, if required, finitely many more exponents for alphabets, languages, dictionaries, thesauruses, and further properties)
An obvious enumeration of the lines is 1, 2, 3, ... where every line n can have many sub-enumerations
n n.1.a n.11.a n.111.a ...
Every section of 1's and a's contains enough symbols to enumerate every single line as often as required to cover all its meanings in all possible languages. If necessary we can add 2's and b's and so on and remain in the countable domain.
This set contains all possible finite words in all possible languages over all possible alphabets over all possible whatever may be required in addition. This set is countable. Hence the set of all definable, computable, and "whatever may be invented by clever set theorists" real numbers is countable as a subset.
Cantor shows that a countable set is uncountable or that a constructed real is unconstructable.
