# Lindström's theorem

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.

## Notes

1. ^ In the sense of Heinz-Dieter Ebbinghaus Extended logics: the general framework in K. J. Barwise and S. Feferman, editors, Model-theoretic logics, 1985 ISBN 0-387-90936-2 page 43
2. ^ A companion to philosophical logic by Dale Jacquette 2005 ISBN 1-4051-4575-7 page 329
3. ^ Chen Chung Chang; H. Jerome Keisler (1990). Model theory. Elsevier. p. 127. ISBN 978-0-444-88054-3.
4. ^ Jean-Yves Béziau (2005). Logica universalis: towards a general theory of logic. Birkhäuser. p. 20. ISBN 978-3-7643-7259-0.
5. ^ Dov M. Gabbay, ed. (1994). What is a logical system?. Clarendon Press. p. 380. ISBN 978-0-19-853859-2.
6. ^ Jouko Väänänen, Lindström's Theorem