bijective maping from natural number set to prime numbers

Question

I think the prime number set is countable . But I am not able to define a bijective mapping from natural number set to prime numbers.

Answer 1

Just define recursively:

$$p_1=2$$ $$p_{n+1}=\text{next prime }>p_n$$ or, more precisely, using the well order property of $\Bbb N$: $$p_{n+1}=\min\{a\in\Bbb N: a>p_n \wedge a \text{ prime}\}$$ There is nothing special here. You can do exactly the same for every infinite subset of $\Bbb N$.