Editing Skolem's paradox (section)

== Background ==

One of the [[Cantor's first set theory article|earliest results]] in [[set theory]], published by Cantor in 1874, was the existence of different sizes, or cardinalities, of infinite sets.{{sfn|Kanamori|1996|p=3}} An infinite set <math> X </math> is called [[countable]] if there is a function that gives a [[one-to-one correspondence]] between <math> X </math> and the [[natural numbers]], and is [[uncountable]] if there is no such correspondence function.<ref>{{Harvnb|Cantor|1874}}. English translation: [[#Ewald|Ewald 1996]], pp. 839{{ndash}}843.</ref>{{sfn|Bays|2007|p=2}} In 1874, Cantor proved that the [[real number]]s were uncountable; in 1891, he proved by his [[Cantor's diagonal argument|diagonal argument]] the more general result known as [[Cantor's theorem]]: for every set <math> S </math>, the [[power set]] of <math> S </math> cannot be in [[bijection]] with <math> S </math> itself.{{sfn|Kanamori|1996|p=7}} When Zermelo proposed [[Zermelo set theory|his axioms for set theory]] in 1908, he proved Cantor's theorem from them to demonstrate their strength.{{sfn|Zermelo|1967|p=200}}

In 1915, [[Leopold Löwenheim]] gave the first proof of what Skolem would prove more generally in 1920 and 1922, the [[Löwenheim–Skolem theorem]].{{sfn|van Heijenoort
|1967|p=232}}{{sfn|Skolem|1967|p=290}} Löwenheim showed that any [[first-order logic|first-order]] sentence with a [[Model theory|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 set|countable]] first-order collection of [[axiomatization|axioms]] is satisfied by an infinite [[structure (mathematical logic)|structure]], then the same axioms are satisfied by some countably infinite structure.{{sfn|Nourani|2014|pp=160{{ndash}}162}} 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.{{sfn|Bays|2007|p=2}}