I am editing the question to express it in more detailed manner. I am actually reading a book of calculus and the countability doubt came to me as a part of reading introduction.
Let me give two problems
Case 1 ::
Set of all natural even numbers are countable. ? Yes we can have a mapping of it to natural numbers.
There is also one more additional facility such that we can say what is the nth element by using the formula f(n) = 2n. This makes sense because we can identify the nth element even it is infinite series
Case 2::
The counting procedure of set of all positive rational numbers is given below(I have changed the previous version before editing to rational number case for clarity ). Here you are cancelling repeated cases like $\frac{2}{2}$ coz $\frac{1}{1}$ is already happened. Yes it is a counting procedure and so it is countable too. But we cant figure out the nth member of it. Basically there is a Counting mechanism but with no nth element calculation.
Question
a) what is the point of such counting method if we cant find the nth element. Could you give an example for the use of such set in mathematics(countable but cant find the nth element). What is the significance of it rather than being just a counting procedure
b) How do we mathematically identify the non countable set rather than trying to make a count sequence by trial and error . An example would be more helpful
Yes, of course! By Schröder-Bernstein we have that $$|\mathbb{Q} \cap [0,1]| \leq |\mathbb{Q}|$$ whith the latter countable. Also it is easy to see that there are infinitely many rationals in the unit interval.