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 le

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.