9. Essentials of Axiomatic method
10. Continuum Hypotheses revisited
11. Unlimited Abstraction Principle and Separation Principle
12. Undecidability of Continuum Hypotheses in Zermelo-Fraenkel system
13. Note on the objections to Zermelo’s system.
How to decide if two sets are of the same size without the appeal to a concept of “size”?
Well, if we could find a way to assign an element of one set to an element of the other set so that each element of one set is assigned to exactly one, unique element of another set and no elements are left over unassigned then these two sets are of the same size. That is, if we can show a one-to-one and onto (i.e., bijective) function between the sets then we know that these two sets have exactly the same number of elements. The concept of being “of the same size” is quite irrelevant here. This method is obviously true for any given finite set of things. Such a finite set can be put into one-to-one correspondence with some subset of integers, say, from 1 to some integer n. This number n is called cardinality of the set.
If the set is infinite, we can attempt to put it into one-to-one correspondence with the set all of integers. If we can show some methodical and well-defined way of doing it (that is, if can show a bijective function between N and a given set A) then we call this set A countable (because we can count it, literally, even if it will take eternity). For example, the set of all even numbers is infinite, but we can define a function f on N such f(n) = 2n. It is a bijective function and so the set of all even integers is of the same size (cardinality) as the set of all integers – countably infinite.