Natural numbers are countable because every one of them already has a name. Think of their unique representation in a given radix system.
This being the case, if any set is in bijection with the set of natural numbers, then it too is countable, that is, we can assign the same names in the set of natural numbers to the other set because the elements are in a one-to-one correspodence. A bijection is NOT required to prove a set is countable. By definition, a set is countable if its elements can be listed, that is, they all have NAMES.
Is the set of "real" numbers countable? Of course not. There are NO real numbers. See linke which proves irrational numbers do not exist, and hence neither can the set of real numbers exist also, because it supposedly includes the non-existent irrational numbers.
Look, I am not going to continue commenting here. I just don't have the time. However, if you read my articles and my comments at STATU, you will learn more than you've ever learned in your entire schooling.
I am a real mathematician. The fools on this forum have accomplished nothing. I , on the other hand have a huge boast in my pocket: I have produced the first and only rigorous formulation of calculus in human history.