Sorry, you missed it. You could formulate the proof with only 32 bit FP numbers. I've created a function that maps any real number, computable or not, to a notional FP representation. The function doesn't actually have to be computable to show a bijection. I just have to show there is a way to do the mapping.
You would have to come up with a mighty strange notion of "floating point representation" that allowed you to have more distinct representations than computable functions
That is what I think I did. Let's say that you have a real number X in [0,1] that you say isn't in the set of possible "floating point representations." I could just posit a reformulation of the IEEE 32 bit FP where NaN is used to represent that number instead. There you have your mapping. Now reset the universe to the state it was 5 minutes ago. In that alternate reality, you say you have a different real number. We also call that real number X and in that other reality I posit a reformulation of the IEEE 32 bit where NaN is used to represent that number X instead.
Just because such a set isn't computable, that doesn't mean it isn't conceivable in a way that can show a bijection.
Even when you move the goalposts (a floating point representation where you can't perform any operations with the numbers you "represent"?) there is still at least one big hole in your trick.
The set of describable real numbers in any particular theory (e.g ZFC) is also countable (because there are a countable number of formulas of any kind), and the set of all formal theories (even including the ones which cannot model the real numbers) is still countable. That's already giving you a lot of leeway with the term "real number". So now you'd have to call your floating point representation a representation of a real number when you can't even specify the X, no matter how abstract or lengthy the description.
In the future if you're going to use your "pedant-card" like that, do it someplace that isn't HN where you'll run headlong into a lot of people who actually know a few basic things about cardinality.
