Infinite essentially mean unknown. How could you possible compare the size of two unknowns?
You just can’t, but we want to be able to reduce something infinite/unknowable, to a finite symbol that we can use in our limited language.
The point is, that's not really what infinite means. A set being infinite just means that we can start listing elements, and we'll always be able to find a new element that we haven't listed before.
To compare two infinite sets A and B, we can consider functions F that take any element from A and output some element from B.
If A and B have the same cardinality, then we can always describe a certain function F that can potentially output any element from B, if we know the right input from A.
But if B has a greater cardinality than A, then we cannot find any such function. No matter which function F we choose, there will always be certain values in B that can never be output by F given any input in A. Thus, we say that B has "more elements" than A in a certain sense.
In the case of Cantor's argument, A is the set of natural numbers, B is the set of real numbers, and F is the ordering we choose.
A variation of Cantor's argument can be expressed fully in terms of finite objects. Suppose we have a function F such that F(i) = f_i, where f_i is a function such that f_i(j) represents the jth digit of the ith real number in the sequence.
Then, we can write a new function g(j) = (F(j)(j) + 2) mod 10, or some variation. This function g(j) similarly represents a real number.
Now, we can take any i we want and start comparing f_i(1) vs. g(1), f_i(2) vs. g(2), etc. After a finite amount of time, we'll always reach a point where the two functions differ. This means that the two functions represent two different real numbers.
Therefore, the real number represented by g is not represented by any of the f_i, no matter which function F we start with.
> A set being infinite just means that we can start listing elements, and we'll always be able to find a new element that we haven't listed before.
You can do that for the real numbers, by keeping a set of all the unique real numbers you’ve produced before, then adding one random real number at a time, always checking that is not in the set already.
And if you could go forever, then the probability of generating every real number is 1.
However, infinity is unknowable. You can’t just say that the “infinite” cardinality of a set can just be reduced to a symbol, that is like saying you finished doing the calculation and that you know there’s a final result. But there isn’t a final result.
The arguments in your explanation only prove that you cannot predict or choose an ordering for the real numbers before counting them. But if you produce random real numbers at each step and re-order them, you don’t need a predefined order. In fact the order can just be the order in which the real number was generated, regardless of its value, then you don’t even need to reorder them.
I don't understand what you're trying to say here. In more finite terms (given that random real numbers already contain infinite information), the diagonalization argument says, "You can keep generating individual real numbers as long as you want, by any process. But there'll always be at least one real number that you'll never ever count to, no matter how long you keep generating numbers."
This does not require any kind of "predefined order". In fact, the diagonalization has to listen to all the numbers we generate just to provide an example of an unreachable number. So at any given point in the process, we can't know in full what the unreachable number is (without knowing the generation process); we only know its initial digits. But the limit of these partial answers is the full unreachable number, which had already existed in the first place.
(I hope you accept the idea of an arithmetic limit; without it, we'd have trouble with basic things like justifying that 0.999... = 1, or resolving Zeno's paradoxes of motion. It's an intrinsic part of what a real number is.)