Without wishing to defend it, the gotcha here is "numbers you name". Rational numbers are "named" by putting together a string of digits. That is not possible for the irrationals so the only ones we can name are a handful with alphabetic names. There are more sheep in a field than those.
Extend alphabetical names to alphabetical descriptions and you have a countable number of irrationals.
It's a bit suspicious that we can't name or describe any of the uncountable irrational numbers, isn't it? Not even a single example.
Cantor's proof is not constructive. It doesn't name an example. If you enumerated all irrationals based on the algorithm required to calculate them then the contradiction in Cantor's diagonalization proof just turns into a failure to terminate. But that suggests that there are fewer irrationals than integers, not more.
You're limiting the existence of real numbers to our (human) capability of mentioning them. To me, that seems a rather arbitrary limit. Did I understand you correctly? If so, why do you accept infinity in the first place?
