Editing Cardinality (section)

==== Skolem's paradox ====
{{Main|Skolem's paradox}}
[[File:Lowenheim-skolem.svg|thumb|Illustration of the [[Löwenheim–Skolem theorem]], where <math>\mathcal{M}</math> and <math>\mathcal{N}</math> are models of set theory, and <math>\kappa</math> is an arbitrary infinite cardinal number.]]
In [[model theory]], a [[Model (mathematical logic)|model]] corresponds to a specific interpretation of a [[formal language]] or [[Theory (mathematical logic)|theory]]. It consists of a [[Domain of discourse|domain]] (a set of objects) and an [[Interpretation (logic)|interpretation]] of the symbols and formulas in the language, such that the axioms of the theory are satisfied within this structure. The [[Löwenheim–Skolem theorem]] shows that any model of set theory in [[first-order logic]], if it is [[consistent]], has an equivalent [[Structure (mathematical logic)|model]] which is countable. This appears contradictory, because [[Georg Cantor]] proved that there exist sets which are not countable. Thus the seeming contradiction is that a model that is itself countable, and which therefore contains only countable sets, [[Satisfiability|satisfies]] the first-order sentence that intuitively states "there are uncountable sets".<ref name=":6">{{Citation |last=Bays |first=Timothy |title=Skolem's Paradox |date=2025 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/paradox-skolem/ |access-date=2025-04-13 |edition=Spring 2025 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri}}</ref>

A mathematical explanation of the paradox, showing that it is not a true contradiction in mathematics, was first given in 1922 by [[Thoralf Skolem]]. He explained that the countability of a set is not absolute, but relative to the model in which the cardinality is measured. Skolem's work was harshly received by [[Ernst Zermelo]], who argued against the limitations of first-order logic and Skolem's notion of "relativity", but the result quickly came to be accepted by the mathematical community.<ref>{{cite journal |last1=van Dalen |first1=Dirk |author-link1=Dirk van Dalen |last2=Ebbinghaus |first2=Heinz-Dieter |author2-link=Heinz-Dieter Ebbinghaus |date=Jun 2000 |title=Zermelo and the Skolem Paradox |url=https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=9d51449b161b9e1d352e103af28fe07f5e15cb4b |journal=The Bulletin of Symbolic Logic |volume=6 |pages=145–161 |citeseerx=10.1.1.137.3354 |doi=10.2307/421203 |jstor=421203 |s2cid=8530810 |number=2 |hdl=1874/27769}}</ref><ref name=":6" />