Good point, and I admit I was arguing the wrong point.
Streams are enumerable, so are lists, maps etc. The difference is that streams are not finite, until they are. That makes streams different from other data structures implementing the enumerable protocol! Thanks for helping me with that 
I gotta be honest, I did not know why exactly it was bugging me so much, but I feel like I am getting a stronger case by gaining some insights.
The set of rationals isn’t, as you can’t clearly map a single natural number to a single rational number.
I think you are confusing rational numbers with real numbers. Rational numbers are perfectly countable, real numbers aren’t
Might be, I’m always confusing both, I’m talking about pi and sqrt(2) etc.
Though I also do not remember the mapping for n/m
natural: 0, 1, 2, 3, …
rational: n/m where n and m both natural numbers
real: e, pi, sqrt(2)
pi and sqrt(2) are irrational numbers, those are indeed uncountable.
Rational numbers are fractions: a/b.
The mapping between natural and rational numbers (proving that they are countable) is actually easier than one might think and kind of cool: https://www.homeschoolmath.net/teaching/rational-numbers-countable.php