Set Theory and The Powerset Function

Abstract: Set theory is the study of the foundations of mathematics, and is rooted in analyzing properties of infinite sets. Historically, the first breakthrough was by Cantor, who proved that the cardinality of the real numbers is strictly bigger that the cardinality of the natural numbers. Namely, there is no bijection between them. More generally, any infinite set has strictly smaller cardinality than its powerset. After Cantor’s theorem, the natural question emerged whether there is something in between. This became known as the Continuum Hypothesis (CH), which states that the answer is no: any infinite subset of the reals is either countable or there is a bijection between it and all the reals.

CH became Hilbert’s First Problem. It was finally resolved by works of Godel (in 1940) and Cohen (in 1964) who showed that CH is independent of the usual mathematical axioms (ZFC). That means that neither CH, not its negation is a logical consequence of the ZFC axioms. Since Cohen’s work, modern set theory investigates ZFC constraints (i.e. “what is necessary”) versus consistency results (i.e. “what is possible”). We will survey these results and then focus on some recent developments in the study of infinite sets.