information:filtering, information:classification, information:extraction, information:highlighting
Background

One of the earliest results in set theory, published by Cantor in 1874, was the existence of different sizes, or cardinalities, of infinite sets. An infinite set X {\displaystyle X} is called countable if there is a function that gives a one-to-one correspondence between X {\displaystyle X} and the natural numbers, and is uncountable if there is no such correspondence function. In 1874, Cantor proved that the real numbers were uncountable; in 1891, he proved by his diagonal argument the more general result known as Cantor's theorem: for every set S {\displaystyle S}, the power set of S {\displaystyle S} cannot be in bijection with S {\displaystyle S} itself. When Zermelo proposed his axioms for set theory in 1908, he proved Cantor's theorem from them to demonstrate their strength.

In 1915, Leopold Löwenheim gave the first proof of what Skolem would prove more generally in 1920 and 1922, the Löwenheim–Skolem theorem. Löwenheim showed that any first-order sentence with a model also has a model with a countable domain; Skolem generalized this to infinite sets of sentences. The downward form of the Löwenheim–Skolem theorem shows that if a countable first-order collection of axioms is satisfied by an infinite structure, then the same axioms are satisfied by some countably infinite structure. Since the first-order versions of standard axioms of set theory (such as Zermelo–Fraenkel set theory) are a countable collection of axioms, this implies that if these axioms are satisfiable, they are satisfiable in some countable model.