The Compactness Theorem
Department of Mathematics
Let us say that a logic has the compactness property if a set of sen- tences of the logic has a model whenever every finite subset of the set has a model. For present purposes, the Compactness Theorem is that first-order logic has the compactness property. This theorem is fundamental to model theory. However, Hodges’s comprehensive twelve-chapter 1993 volume Model Theory finds no need to state and prove the theorem until Chapter 6. It is worthwhile to think about what needs Compactness and what does not.
One consequence of the Compactness Theorem is that a set of sen- tences with arbitrarily large finite models must have an infinite model. A more purely mathematical consequence is the Prime Ideal Theorem: every nontrivial commutative ring has a prime ideal. One can prove this by noting first that every maximal ideal is prime. Moreover, every countable ring has a maximal ideal; for we can obtain a generating set of such an ideal by considering the elements of the ring one by one. In particular then, every finitely generated subring of a given ring has a maximal ideal, because every finitely generated ring is countable. By the Compactness Theorem then, the original ring must have an ideal that is at least prime, although it might not be maximal. The point here is that primeness is a “local” property, while maximality is not.
It is usually understood that every nontrivial commutative ring has, not just a prime ideal, but a maximal ideal. To make it easy to prove such results, Zorn stated in 1935 the result now known by his name. However, Zorn’s Lemma relies on the Axiom of Choice. The Com- pactness Theorem is strictly weaker than this, with respect to ZF (Zermelo–Fraenkel set theory without Choice). For, Compactness is also a consequence of the Prime Ideal Theorem, even the Boolean Prime Ideal Theorem; and this is strictly weaker than the Axiom of Choice (as shown by Halpern and Lévy in 1971).
The Compactness Theorem for countable sets of sentences needs nothing beyond ZF. Skolem showed this implicitly in 1922 when he established the paradox that Zermelo’s axioms for set theory must have a countable model, if they have a model at all. In 1930, Gödel proved countable Compactness explicitly, though not by that name. Mal'tsev stated the full Compactness Theorem as the General Local Theorem in 1941, having proved it implicitly in 1936; he used it to prove algebraic results in the way we proved the Prime Ideal Theorem above.
In his 1950 address to the International Congress of Mathemati- cians, Tarski gave the Compactness Theorem its current name and noted its topological meaning. But this meaning is not generally well expressed in today’s textbooks of model theory.
The class of structures having a given signature can be given a topol- ogy, although the closed “sets” in this topology are not sets, but proper classes (except for the empty set): they are the classes of models of sets of sentences. The space of all structures has a Kol- mogorov (T0 ) quotient that is a set: it is the space of complete theories of structures. If one replaces sentences with their logical equivalence classes, then the set of sentences becomes a Boolean algebra, called a Lindenbaum algebra; and the complete theories of structures become ultrafilters of the Lindenbaum algebra. By means of the Boolean Prime Ideal Theorem, the Stone space consisting of all ultrafilters of the Lindenbaum algebra is easily shown to be com- pact. Or one could look instead at the spectrum, consisting of the prime ideals of the corresponding Boolean ring; the spectrum of ev- ery ring is compact. The Compactness Theorem says more: every ultrafilter of the Lindenbaum algebra is derived from the complete theory of a structure.
The compactness theorem for propositional logic can be seen as a version of the theorem that the product of two-element discrete spaces (or indeed any compact Hausdorff spaces) is compact. The Compactness Theorem for first-order logic does not follow so readily, though it can be seen to result from a kind of reduction of first-order logic to propositional logic. Then Lindström’s Theorem is roughly that there is no such reduction for certain more expressive logics—but see that tutorial for more. Sometimes the Compactness Theorem is derived from the Completeness Theorem: see that tutorial for more. Meanwhile, the present tutorial is intended to fill out the foregoing sketch of the Compactness Theorem as such.
 JohnW. Dawson, Jr. The compactness of first-order logic: from Gödel to Lindström. Hist. Philos. Logic, 14(1):15–37, 1993.
 Kurt Gödel. The completeness of the axioms of the functional calculus of logic. In van Heijenoort , pages 582–91. First published 1930.
 J. D. Halpern and A. Lévy. The Boolean prime ideal theorem does not imply the axiom of choice. In Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif., 1967), pages 83–134. Amer. Math. Soc., Providence, R.I., 1971.
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