Intuition Regarding Countability of Infinite Sets and Infinite Sets as "Whole Objects" I have some issue wrapping my mind around the concepts of enumeration of infinite sets, and infinite sets as whole complete objects in general. 
I know a lot of people struggle with this, and some people (even some of the greatest mathematicians of all time) have resisted these notions. I just want to get past the point where I’m haunted by ambiguities that I perceive through standard explanations. 
The first thing I want to clear up is the notion that an infinite set can be enumerated. I know that we say that an infinite set is enumerable if it can be put into bijection with the natural numbers. This makes sense from the surface, but my senses fail me at the part where we assume that such a set can actually be “enumerated” in the sense of a verb, i.e. enumeration is an action that on an infinite set can be completed. 
My intuition suggests that such a definition is actually only a rule for enumeration, but there is no such thing as a completed enumeration. The idea being that you “can describe how to do something that cannot be done".
With this problem I have a problem conceptualizing Cantors diagonalization proof. It goes by assuming something is an enumeration of some infinite set. Then it selects something not in that set by selecting something that differs by value for each digit with respect to every element in the enumerated set. It seams like you first need the notion that selecting this new element can ever be done, since there is no end (no base case where such a procedure would ever end). Such an element seams "un-selectable" by nature purely by assumption that the supposedly enumerated set is infinite. So to me it seams like trying to select this element is a little like trying to select the largest natural number.
Maybe I am coming at this from the wrong angle. Any insight is appreciated. 
 A: The key is to stop thinking about enumerations as something that you do, and think about them as something that exists in the mathematical universe.
Of course, asking you to think that something exists does not make it exist in general. I could think of an even number less than $2$, it won't make it pop into mathematical existence, and I can think about a cold pint of beer right now, but unless I go outside and buy myself a cold pint of beer, it doesn't exist in my current world.
So how do we bridge that issue? We are handed a mathematical universe with the tacit agreement that it satisfies some axioms. These could be set theory axioms, or some type theory, or other foundational theory which gives us a general framework. This universe is given to you, with everything inside of it, and now the question is what can you ensure that is there if you use the axioms and the inference rules.
Now the definition of being countable makes more sense. If there is a function from the natural numbers to some set which is a bijection, then the set is countable, and that function is call an enumeration. We can prove that the rational numbers are countable, and we can prove that some sets are countable; and Cantor's diagonal argument shows that no matter what, we cannot prove that the real numbers are countable.
Namely, the idea is that given any function from the natural numbers into the real numbers, we can define from that function a real number which is not any of the values given by our function. But the function is inside our universe, and the tacit axioms of our universe should allow us to use definitions like that.
It is true, however, that in another universe the diagonal proof might not work. In some constructive settings it can fail, or if you assume there are no such thing as irrational numbers, then it will fail.
But in general, most of mathematics is not concerned with you enumerating, but rather with you proving the existence or nonexistence of an enumerating function.
A: An infinite set is countable if there exists a perfect scheme to count the elements of the set irrespective of actually completing the process of counting. The idea is rather, focusing on not to miss any element during the counting. I think this is what we call an 'optimistic approach' who thinks that if you are not missing anything during counting, you would certainly be able to count the whole set some day.
