In mathematical logic, Lindström's theorem (named after Swedish logician Per Lindström, who published it in 1969) states that first-order logic is the strongest logic^[1] (satisfying certain conditions, e.g. closure under classical negation) having both the (countable) compactness property and the (downward) Löwenheim–Skolem property.^[2]

Lindström's theorem is perhaps the best known result of what later became known as abstract model theory,^[3] the basic notion of which is an abstract logic;^[4] the more general notion of an institution was later introduced, which advances from a set-theoretical notion of model to a category-theoretical one.^[5] Lindström had previously obtained a similar result in studying first-order logics extended with Lindström quantifiers.^[6]

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

• Per Lindström, "On Extensions of Elementary Logic", Theoria 35, 1969, 1–11. doi:10.1111/j.1755-2567.1969.tb00356.x
• Johan van Benthem, "A New Modal Lindström Theorem", Logica Universalis 1, 2007, 125–128. doi:10.1007/s11787-006-0006-3
• Ebbinghaus, Heinz-Dieter; Flum, Jörg; Thomas, Wolfgang (1994), Mathematical Logic (2nd ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-387-94258-2
• Sebastian Enqvist, "A General Lindström Theorem for Some Normal Modal Logics", Logica Universalis 7, 2013, 233–264. doi:10.1007/s11787-013-0078-9
• Monk, J. Donald (1976), Mathematical Logic, Graduate Texts in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90170-1
• Shawn Hedman, A first course in logic: an introduction to model theory, proof theory, computability, and complexity, Oxford University Press, 2004, ISBN 0-19-852981-3, section 9.4