I was trying to understand the undecidable nature of the continuum hypothesis and came up with the following question:
The set of circles with a rational diameter is countably infinite (with cardinality equal to the cardinality of integers). The set of circles with a rational circumference is countably infinite (with cardinality equal to the cardinality of integers). The cardinality of the union of these sets is clearly smaller than the uncountable set of irrational numbers, but why isn't it larger than the set of countable integers?