mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...

, Lindström's theorem (named after Swedish logician Per Lindström, who published it in 1969) states that

first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quanti ...

is the '' strongest logic'' (satisfying certain conditions, e.g. closure under classical negation) having both the (countable) compactness property and the (downward) Löwenheim–Skolem property. Lindström's theorem is perhaps the best known result of what later became known as abstract model theory, the basic notion of which is an

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; the more general notion of an institution was later introduced, which advances from a set-theoretical notion of model to a

category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) ...

-theoretical one. Lindström had previously obtained a similar result in studying first-order logics extended with

Lindström quantifier In mathematical logic, a Lindström quantifier is a generalized polyadic quantifier. Lindström quantifiers generalize first-order quantifiers, such as the existential quantifier, the universal quantifier, and the counting quantifiers. They were i ...

Jouko Väänänen Jouko Antero Väänänen (born September 3, 1950 in Rovaniemi, Lapland) is a Finnish mathematical logician known for his contributions to set theory,J. VäänänenSecond order logic or set theory? Bulletin of Symbolic Logic, 18(1), 91-121, 2012 ...

Lindström's Theorem
/ref> Lindström's theorem has been extended to various other systems of logic, in particular modal logics by Johan van Benthem and Sebastian Enqvist.

Notes

References

* Per Lindström, "On Extensions of Elementary Logic", ''

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'' 35, 1969, 1–11. * Johan van Benthem, "A New Modal Lindström Theorem", ''

Logica Universalis ''Logica Universalis'' is a peer-reviewed Peer review is the evaluation of work by one or more people with similar competencies as the producers of the work (peers). It functions as a form of self-regulation by qualified members of a profess ...

'' 1, 2007, 125–128. * * Sebastian Enqvist, "A General Lindström Theorem for Some Normal Modal Logics", ''

'' 7, 2013, 233–264. * * Shawn Hedman, ''A first course in logic: an introduction to model theory, proof theory, computability, and complexity'', Oxford University Press, 2004, , section 9.4 {{Mathlogic-stub Mathematical logic Theorems in the foundations of mathematics Metatheorems