It is not a surprise that the number of integers (1,2 3, 4, etc…) has an infinite number of numbers. The number of irrational numbers is also infinite but it is a larger number of infinite numbers. And any list of numbers has a ‘Power Set’ which consists of all possible combination of numbers in that list. That power set will always have more numbers than the original list so the power set of irrational numbers is a larger infinite number than the original list. And you can then take the power set of the power set and so on. This means that there are an infinite number of different sizes of infinite numbers.

(The Great Courses – Zero to Infinity: A History of Numbers)