[powera]

I haven't the slightest idea what "n" and "m" are supposed to be, so I can't make any sense of your definition.

Beyond that, you say: "all real numbers that you're likely to hear of" - this is a fallacy twice over.

First, since my lifetime is finite, it's easy to count all the real numbers that I will hear in my life. But I don't think that's the argument you want to make.

Second, there are people who professionally construct unlikely real numbers; just because you choose to ignore those numbers doesn't mean that your countable sequence of all real numbers can. There's a difference between "things that cannot be written down" and "things you can't write down".

[btilly]

The statement "you are likely to hear of" is a concession that I know of numbers which can't fit that definition. Their key characteristic being that they cannot be calculated by humans, because evaluating them requires operations that are not actually possible for us.

Whether or not such numbers are well-defined at all is a point of philosophy. What does it mean to declare the truth of statements that cannot be verified either way?