I understand that irrational numbers are uncountable. I’ve seen the proof and it makes perfect sense. However, I came up with this (most likely false) proof that says that they’re countable. Chances are, I made a mistake and the proof doesn’t mean anything, but I just want to be sure. Here is the proof:

You can’t find a solid chunk (range of numbers) that does not contain a rational number. Which means that irrational numbers are just points on the number line, not lines. Which means you just need to name all the points to count the irrationals.

What is the mistake here?