Advanced Mathematics

2.6Larger Infinities and the ZFC Axioms

Purpose of Section

We state and prove Cantor's power set theorem, which guarantees the existence of infinities larger than the cardinality of the continuum. We also state and discuss the Zermelo–Fraenkel axioms, which are the most commonly used system of axioms for set theory, as well as the Axiom of Choice (AC) and the Continuum Hypothesis (CH), topics that lie at the center of the foundation of set theory and mathematics.

2.6.1 Cantor's Discovery of Larger Sets

The nineteenth‐century German mathematician Georg Cantor must have felt that he was on a great adventure. He had just discovered that there were two kinds of infinity, the counting infinity ℵ₀ of the natural numbers and the larger continuum infinity c of the real numbers. He then wondered if there were still more infinities; infinities even larger than that of c. He also wondered if there was an infinity between ℵ₀ and c, or in other words, is there a set whose cardinality is greater than the cardinality ℵ₀ of the natural numbers and less than the cardinality c of the real numbers. He would spend the remainder of his life trying to answer that question and the result will amaze you.

Little ideas often lead to big ideas. We have seen that for finite sets, the power set of a set is larger than the set itself. For example, the set A = {a, b, c} contains three elements, whereas its power set

has 2³ = 8 elements. This prompted Cantor to ask if the same property held for infinite ...