Skolem's paradox

In <a href="/facts/Mathematical_logic/SaI7Y3wN">mathematical logic</a> and <a href="/facts/Philosophy/zAH5PVD6">philosophy</a>, Skolem's paradox is the apparent contradiction that a <a href="/facts/Countable_set/Wj4DmWPv">countable</a> <a href="/facts/Model_theory/vLWVV8sQ">model</a> of <a href="/facts/First-order_logic/lcISqWIg">first-order</a> <a href="/facts/Set_theory/1ptfsvUP">set theory</a> could contain an <a href="/facts/Uncountable/zl2WqyJQ">uncountable</a> <a href="/facts/Set_(mathematics)/BfucfHMq">set</a>. The paradox arises from part of the <a href="/facts/L%C3%B6wenheim%E2%80%93Skolem_theorem/pb72J40H">Löwenheim–Skolem theorem</a>; <a href="/facts/Thoralf_Skolem/M7t2mfGx">Thoralf Skolem</a> was the first to discuss the seemingly contradictory aspects of the theorem, and to discover the relativity of set-theoretic notions now known as non-<a href="/facts/Absoluteness_(mathematical_logic)/rNkAooUb">absoluteness</a>. Although it is not an actual <a href="/facts/Antinomy/6pTJIYfX">antinomy</a> like <a href="/facts/Russell%27s_paradox/XupqWGYt">Russell's paradox</a>, the result is typically called a <a href="/facts/Paradox/3VLGjuWf">paradox</a> and was described as a "paradoxical state of affairs" by Skolem.
In model theory, a model corresponds to a specific interpretation of a <a href="/facts/Formal_language/crDTyP8q">formal language</a> or theory. It consists of a domain (a set of objects) and an 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 <a href="/facts/First-order_logic/lcISqWIg">first-order logic</a>, if it is <a href="/facts/Consistent/IHkdnA9M">consistent</a>, has an equivalent <a href="/facts/Structure_(mathematical_logic)/GsMx1fEb">model</a> that is countable. This appears contradictory, because <a href="/facts/Georg_Cantor/k5v1DvW4">Georg Cantor</a> proved that there exist sets which are <a href="/facts/Cantor%27s_diagonal_argument/tDyYAN05">not countable</a>. Thus the seeming contradiction is that a model that is itself countable, and which therefore contains only countable sets, <a href="/facts/Satisfiability/NkmQFgHB">satisfies</a> the first-order sentence that intuitively states "there are uncountable sets".
A mathematical explanation of the paradox, showing that it is not a true contradiction in mathematics, was first given in 1922 by Skolem. He explained that the countability of a set is not absolute, but relative to the model in which the <a href="/facts/Cardinality/m6VPojwT">cardinality</a> is measured. Skolem's work was harshly received by <a href="/facts/Ernst_Zermelo/uvW8gDq0">Ernst Zermelo</a>, 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.
The philosophical implications of Skolem's paradox have received much study. One line of inquiry questions whether it is accurate to claim that any first-order sentence actually states "there are uncountable sets". This line of thought can be extended to question whether any set is uncountable in an absolute sense. More recently, scholars such as <a href="/facts/Hilary_Putnam/ujwAh4Ry">Hilary Putnam</a> have introduced the paradox and Skolem's concept of relativity to the study of the <a href="/facts/Philosophy_of_language/qwNCwF4W">philosophy of language</a>.

Skolem's paradox open-in-new

Skolem's paradox