I'm on your side for most of what you say. This topic has been interesting to me for years. I've considered going back to school to build on my math degree, specifically because of this topic.

However, I thought things like Chaitin's Constants (you could make one per programming language) are real numbers you can name but not compute. I think you could do this from any undecidable problem.

Of course there only a countable number of those Reals. And they still don't seem useful for much more than naval gazing.