If irrational numbers are uncountable, then why did I find this? 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?

"You can't find a solid chunk (range of numbers) that does not contain a rational number." - True, this is just the statements that the rationals are dense in the reals (and also irrationals).
"Which means that irrational numbers are just points on the number line, not lines." - True
"Which means you just need to name all the points to count the irrationals." - Basically saying "we need to give a name to everything in this bucket".  As @Deepak noted, this is the hard part.  Note that whatever (finite!) "name" you come up with, it can be mapped to the naturals via eg dictionary sorting.  This is basically the statement that the computable irrationals are still countable - whether "sqrt of pi" or "8th root of polynomial: ___".  Beyond these are an uncountable sea of irrationals that cannot be computed/described/etc by any finite means.  How do you plan to name these?
