Cantorian set theory

Cantorian set theory, founded by Georg Cantor, explores the concept of sets and their sizes, known as cardinalities. It establishes that even among infinite sets, there are different sizes of infinity. For instance, the set of rational numbers is infinite but countable, meaning it can be put into a one-to-one correspondence with the natural numbers. However, the set of real numbers is a larger, uncountable infinity.123

Cantor's theorem states that the cardinality of any set is strictly less than the cardinality of its power set (the set of all its subsets). This led to the concept of transfinite numbers, which are used to measure the sizes of infinite sets. The theory also introduced the continuum hypothesis, which posits there are no cardinal numbers between the infinity of the integers and the infinity of the real numbers.142