# Circles and the continuum hypothesis

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?

• Union of two countable is countable. – André Nicolas Sep 11 '14 at 1:13
• No, the union of two countable sets is also countable. – Epsilon Sep 11 '14 at 1:13
• So the same could be said about the set of right triangles whose base and height are rational? – Chris L. Sep 11 '14 at 1:26
• @user1688944 yes: there are countably many of those, too. – Omnomnomnom Sep 11 '14 at 1:34
• No, the set of circles with a rational diameter is not countable. In fact, there are uncountably many circles with diameter one. – bof Sep 11 '14 at 6:23