Skolem's paradox

Skolem's paradox is the apparent tension that first-order set theory has a countable model containing sets the model regards as uncountable. It arises because uncountability is expressed internally using the model's own objects and functions.

First-order set theory: Set theory axiomatized in first-order logic, typically with membership as its sole nonlogical relation. Its first-order language cannot quantify over every subset or function existing outside a model.
Model theory: The mathematical study of structures that satisfy formal languages and theories. Skolem's paradox is a consequence of how first-order theories admit models.
Thoralf Skolem: Norwegian mathematician whose work helped establish modern model theory and the relative character of set-theoretic notions. His 1922 discussion made the countable-model tension prominent.
Cantor's theorem: The theorem that no set can be put in bijection with its power set. Its internal proof remains valid in the model, even when the model's domain is externally countable.
Countable model: A mathematical structure whose domain has a bijection with the natural numbers. The paradox concerns a model whose domain is countable from an external viewpoint.
First-order logic: A formal logic whose quantifiers range over individual objects in a domain. Its quantifiers range over the model's domain, not over external collections of subsets.
Löwenheim–Skolem theorem: A model-theoretic theorem stating that first-order theories with infinite models have models of different infinite cardinalities under specified conditions. The theorem supplies the countable model that makes the apparent paradox possible.
Categorical theory: A theory whose models are all isomorphic within a specified cardinality. First-order set theory lacks categoricity across infinite cardinalities, allowing models of varied sizes.
Internal model: A structure interpreted from within another structure, whose objects and relations are evaluated using the larger structure's resources. The model's own definitions determine which functions count as bijections.
Structure (mathematical logic): A domain equipped with interpretations of the symbols of a formal language. A set-theoretic model is a structure with a domain and an interpretation of membership.

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