Countable Set or not

Question

How to know whether a enumeration method exist or not for a given set to prove it countable?? For example how to know that set of irrational numbers is countable or not..

Comments

To check whether enumaration technique exists or not we have to correlate each element of set with one to one relationship with natural numbers.

Eg. Set of Even numbers

E = {0,2,4,6, 8, .....} then we can correlate natural numbers N = {1,2, 3, 4,...} to each element of E.

such as. E = 2i , N = i+1 | i>=0

But we can't do the same with set of irrational numbers.

Let irrational number is 3.145635.... (no repetition) then we can't give any generalized enumartion to represent these irrational number in the form of natural numbers.

Hence irrational numbers are not countable.