In
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 quantifie ...
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
abstract logic;
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 quantifiers.
[ Jouko Väänänen]
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", ''Theoria
Christian mysticism is the tradition of mystical practices and mystical theology within Christianity which "concerns the preparation f the personfor, the consciousness of, and the effect of ..a direct and transformative presence of God" ...
'' 35, 1969, 1–11.
* Johan van Benthem, "A New Modal Lindström Theorem", '' Logica Universalis'' 1, 2007, 125–128.
*
* Sebastian Enqvist, "A General Lindström Theorem for Some Normal Modal Logics", '' Logica Universalis'' 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