You need to decide whether you want to define counting as “there being a next element”, or “being able to produce those elements in a finite way with a predefined order”.
The issue here is that you are trying to make a finite/determinate assertion about an infinite process.
There are no reals in between 0.001 and 0.002, unless you generate them. And when you do, you can count them one by one until you finish generating them.
Just like you could never finish counting infinite real numbers, you could never finish counting natural numbers.
You seem to be thinking of a real as a string of digits, or a block of bits, which is just the representation of a real. Like, a number doesn't exist until you think of it. That view doesn't help much in thinking about these matters.
> So then what Cantors proof is saying is that if you could somehow count to infinity and get to a result, some infinities would have more elements than others. But that is a contradiction, because you can’t finish counting to infinity.
So you're saying Cantor's "proof" embodies a contradiction. You must be a very great mathematician. I'm not a mathematician, so I'd better just bow out now. Anyway, I'm way too old to start thinking of numbers as some kind of generative process.
That’s just gate keeping through labels. And probably the reason why so many old ideas go completely unchecked and are taken as gospel for so long with no one questioning them.
You can clearly reason about this stuff, even if you are not a mathematician. But maybe it scares you to think that something that you thought was “true”, is just someone else’s opinion about what they thought it was true.
> Simple: at each step generate a random real number and assign it a natural number.
This is nonsense. You said you were going to go through the reals "in order". Then you give a procedure that involves generating reals randomly. Just think about what you said.
It's not possible to go through the reals "in order"; that would imply that for any real, there is a "next" real.
Let the starting real be A. Suppose there is a next real; let it be B. The interval between them, B-A, let us call that C. I can divide C by 2, and produce a real A+(C/2), which is smaller than B and larger than A. Therefore B is not the next real.
This is true for any choice of A and B, so it follows that there cannot be a "next" real, so it's impossible to go through the reals in order.