Paradoxes of set theory

This article contains a discussion of paradoxes of set theory. As with most mathematical paradoxes, they generally reveal surprising and counter-intuitive mathematical results, rather than actual logical contradictions within modern axiomatic set theory.

Basics

Cardinal numbers

Set theory as conceived by Georg Cantor assumes the existence of infinite sets. As this assumption cannot be proved from first principles it has been introduced into axiomatic set theory by the axiom of infinity, which asserts the existence of the set N of natural numbers. Every infinite set which can be enumerated by natural numbers is the same size (cardinality) as N, and is said to be countable. Examples of countably infinite sets are the natural numbers, the even numbers, the prime numbers, and also all the rational numbers, i.e., the fractions. These sets have in common the cardinal number |N| =

ℵ

0

{\displaystyle \aleph _{0}}

(aleph-nought), a number greater than every natural number.

Cardinal numbers can be defined as follows. Define two sets to have the same size by: there exists a bijection between the two sets (a one-to-one correspondence between the elements). Then a cardinal number is, by definition, a class consisting of all sets of the same size.