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

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
A companion to philosophical logic by Dale Jacquette 2005 ISBN 1-4051-4575-7 page 329
Chen Chung Chang; H. Jerome Keisler (1990). Model theory. Elsevier. p. 127. ISBN 978-0-444-88054-3.
Jean-Yves Béziau (2005). Logica universalis: towards a general theory of logic. Birkhäuser. p. 20. ISBN 978-3-7643-7259-0.
Dov M. Gabbay, ed. (1994). What is a logical system?. Clarendon Press. p. 380. ISBN 978-0-19-853859-2.
Jouko Väänänen, Lindström's Theorem

References

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

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[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

[ChangKeisler1990-3] Chen Chung Chang; H. Jerome Keisler (1990). Model theory. Elsevier. p. 127. ISBN 978-0-444-88054-3.

[Béziau2005-4] Jean-Yves Béziau (2005). Logica universalis: towards a general theory of logic. Birkhäuser. p. 20. ISBN 978-3-7643-7259-0.

[Gabbay1994-5] 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