Answer to Question #279511 in Math for Yangyang

Question

... ℵ0 is the least transfinite cardinal number. If a is any transfinite cardinal number different from ℵ0, then

ℵ0 < a.

Exercise 2.49. Prove the above claim of Cantor. (Hint: Let A be an infinite set and let a = |A|. Can you define a one-to-one function from N to A? What can you conclude about the relationship between ℵ0 and a? To define such a function you will need to make countably-many choices of some elements of A. The ability to make such choices depends on what is known as the Axiom of Choice — an important but rather controversial principle in mathematics. A detailed discussion of this axiom and its history can be found in [1] and in the references therein.)