>>3
X is countable if there is an injective function from X to the set of natural numbers.
>>5
You're using axiom of choice. Without axiom of choice you don't have the "countable union of countable sets is countable" theorem. Also without axiom of choice, suprajective functions doesn't always have inverse.
>>6
an algebraic number is a complex number that is a root of a non-zero polynomial in one variable with rational coefficients.