Real numbers and countability

A set is said to be countable if there is bijection between a subset of natural numbers (or even integers) and that set.

http://en.wikipedia.org/wiki/Countable_set

Cantor's diagonalization argument shows that the set of reals is uncountable. So, there are uncountably many subsets of real numbers that are countable. For example the set consisting of any single real number.

Computability has nothing to do with counting the number of elements in a set. This is related to the idea of computable functions.

http://en.wikipedia.org/wiki/Computable_function

A real number is computable if there is finite algorithm that can yield it's digits to any desired level of accuracy.

It is false, because you can always pick a countable subset of your choice (e.g. {1, 2}). A set is countable if it can be put into one-to-one correspondence with the set of Natural numbers or some subset of Natural numbers. Since Natural numbers are contained in Real numbers, you can always pick a subset (like the one mentioned above) that would be countable.

Taking the general definition of a subset, the answer is false. $\mathbb{N}$ itself is a subset of $\mathbb{R}$, and it is countable by definition--in fact, by taking multiples of 2, 3, 4, and so on, you could get infinitely many countable subsets of $\mathbb{R}$. There are also finite subsets of $\mathbb{R}$ (like $\{\pi\}$ or $\{1\}$).

If you are thinking of intervals from $\mathbb{R}$ like $(0,1)$ or $(5,\infty)$, then there are no countable non-empty intervals, since you can always create a bijection between an open interval and the whole real line.