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.