Compactness theorem

Compactness theorem In mathematical logic, the compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful (but generally not effective) method for constructing models of any set of sentences that is finitely consistent.

The compactness theorem for the propositional calculus is a consequence of Tychonoff's theorem (which says that the product of compact spaces is compact) applied to compact Stone spaces,[1] hence the theorem's name. Likewise, it is analogous to the finite intersection property characterization of compactness in topological spaces: a collection of closed sets in a compact space has a non-empty intersection if every finite subcollection has a non-empty intersection.

The compactness theorem is one of the two key properties, along with the downward Löwenheim–Skolem theorem, that is used in Lindström's theorem to characterize first-order logic. Although, there are some generalizations of the compactness theorem to non-first-order logics, the compactness theorem itself does not hold in them, except for a very limited number of examples.[2] Contents 1 History 2 Applications 2.1 Robinson's principle 2.2 Upward Löwenheim–Skolem theorem 2.3 Non-standard analysis 3 Proofs 4 See also 5 Notes 6 References 7 External links History Kurt Gödel proved the countable compactness theorem in 1930. Anatoly Maltsev proved the uncountable case in 1936.[3][4] Applications The compactness theorem has many applications in model theory; a few typical results are sketched here.

Robinson's principle The compactness theorem implies the following result, stated by Abraham Robinson in his 1949 dissertation.

Robinson's principle:[5][6] If a first-order sentence holds in every field of characteristic zero, then there exists a constant {displaystyle p} such that the sentence holds for every field of characteristic larger than {displaystyle p.} This can be seen as follows: suppose {displaystyle varphi } is a sentence that holds in every field of characteristic zero. Then its negation {displaystyle lnot varphi ,} together with the field axioms and the infinite sequence of sentences {displaystyle 1+1neq 0,;;1+1+1neq 0,;ldots } is not satisfiable (because there is no field of characteristic 0 in which {displaystyle lnot varphi } holds, and the infinite sequence of sentences ensures any model would be a field of characteristic 0). Therefore, there is a finite subset {displaystyle A} of these sentences that is not satisfiable. {displaystyle A} must contain {displaystyle lnot varphi } because otherwise it would be satisfiable. Because adding more sentences to {displaystyle A} does not change unsatisfiability, we can assume that {displaystyle A} contains the field axioms and, for some {displaystyle k,} the first {displaystyle k} sentences of the form {displaystyle 1+1+cdots +1neq 0.} Let {displaystyle B} contain all the sentences of {displaystyle A} except {displaystyle lnot varphi .} Then any field with a characteristic greater than {displaystyle k} is a model of {displaystyle B,} and {displaystyle lnot varphi } together with {displaystyle B} is not satisfiable. This means that {displaystyle varphi } must hold in every model of {displaystyle B,} which means precisely that {displaystyle varphi } holds in every field of characteristic greater than {displaystyle k.} This completes the proof.

The Lefschetz principle, one of the first examples of a transfer principle, extends this result. A first-order sentence {displaystyle varphi } in the language of rings is true in some (or equivalently, in every) algebraically closed field of characteristic 0 (such as the complex numbers for instance) if and only if there exist infinitely many primes {displaystyle p} for which {displaystyle varphi } is true in some algebraically closed field of characteristic {displaystyle p,} in which case {displaystyle varphi } is true in all algebraically closed fields of sufficiently large non-0 characteristic {displaystyle p.} [5] One consequence is the following special case of the Ax–Grothendieck theorem: all injective complex polynomials {displaystyle mathbb {C} ^{n}to mathbb {C} ^{n}} are surjective[5] (indeed, it can even be shown that its inverse will also be a polynomial).[7] In fact, the surjectivity conclusion remains true for any injective polynomial {displaystyle F^{n}to F^{n}} where {displaystyle F} is a finite field or the algebraic closure of such a field.[7] Upward Löwenheim–Skolem theorem A second application of the compactness theorem shows that any theory that has arbitrarily large finite models, or a single infinite model, has models of arbitrary large cardinality (this is the Upward Löwenheim–Skolem theorem). So for instance, there are nonstandard models of Peano arithmetic with uncountably many 'natural numbers'. To achieve this, let {displaystyle T} be the initial theory and let {displaystyle kappa } be any cardinal number. Add to the language of {displaystyle T} one constant symbol for every element of {displaystyle kappa .} Then add to {displaystyle T} a collection of sentences that say that the objects denoted by any two distinct constant symbols from the new collection are distinct (this is a collection of {displaystyle kappa ^{2}} sentences). Since every finite subset of this new theory is satisfiable by a sufficiently large finite model of {displaystyle T,} or by any infinite model, the entire extended theory is satisfiable. But any model of the extended theory has cardinality at least {displaystyle kappa } Non-standard analysis A third application of the compactness theorem is the construction of nonstandard models of the real numbers, that is, consistent extensions of the theory of the real numbers that contain "infinitesimal" numbers. To see this, let {displaystyle Sigma } be a first-order axiomatization of the theory of the real numbers. Consider the theory obtained by adding a new constant symbol {displaystyle varepsilon } to the language and adjoining to {displaystyle Sigma } the axiom {displaystyle varepsilon >0} and the axioms {displaystyle varepsilon <{tfrac {1}{n}}} for all positive integers {displaystyle n.} Clearly, the standard real numbers {displaystyle mathbb {R} } are a model for every finite subset of these axioms, because the real numbers satisfy everything in {displaystyle Sigma } and, by suitable choice of {displaystyle varepsilon ,} can be made to satisfy any finite subset of the axioms about {displaystyle varepsilon .} By the compactness theorem, there is a model {displaystyle {}^{*}mathbb {R} } that satisfies {displaystyle Sigma } and also contains an infinitesimal element {displaystyle varepsilon .} A similar argument, adjoining axioms {displaystyle omega >0,;omega >1,ldots ,} etc., shows that the existence of infinitely large integers cannot be ruled out by any axiomatization {displaystyle Sigma } of the reals.[8] Proofs One can prove the compactness theorem using Gödel's completeness theorem, which establishes that a set of sentences is satisfiable if and only if no contradiction can be proven from it. Since proofs are always finite and therefore involve only finitely many of the given sentences, the compactness theorem follows. In fact, the compactness theorem is equivalent to Gödel's completeness theorem, and both are equivalent to the Boolean prime ideal theorem, a weak form of the axiom of choice.[9] Gödel originally proved the compactness theorem in just this way, but later some "purely semantic" proofs of the compactness theorem were found; that is, proofs that refer to truth but not to provability. One of those proofs relies on ultraproducts hinging on the axiom of choice as follows: Proof: Fix a first-order language {displaystyle L,} and let {displaystyle Sigma } be a collection of L-sentences such that every finite subcollection of {displaystyle L} -sentences, {displaystyle isubseteq Sigma } of it has a model {displaystyle {mathcal {M}}_{i}.} Also let {textstyle prod _{isubseteq Sigma }{mathcal {M}}_{i}} be the direct product of the structures and {displaystyle I} be the collection of finite subsets of {displaystyle Sigma .} For each {displaystyle iin I,} let {displaystyle A_{i}={jin I:jsupseteq i}.} The family of all of these sets {displaystyle A_{i}} generates a proper filter, so there is an ultrafilter {displaystyle U} containing all sets of the form {displaystyle A_{i}.} Now for any formula {displaystyle varphi } in {displaystyle Sigma :} the set {displaystyle A_{{varphi }}} is in {displaystyle U} whenever {displaystyle jin A_{{varphi }},} then {displaystyle varphi in j,} hence {displaystyle varphi } holds in {displaystyle {mathcal {M}}_{j}} the set of all {displaystyle j} with the property that {displaystyle varphi } holds in {displaystyle {mathcal {M}}_{j}} is a superset of {displaystyle A_{{varphi }},} hence also in {displaystyle U} Łoś's theorem now implies that {displaystyle varphi } holds in the ultraproduct {textstyle prod _{isubseteq Sigma }{mathcal {M}}_{i}/U.} So this ultraproduct satisfies all formulas in {displaystyle Sigma .} See also Barwise compactness theorem Herbrand's theorem List of Boolean algebra topics Löwenheim–Skolem theorem – Existence and cardinality of models of logical theories Notes ^ See Truss (1997). ^ J. Barwise, S. Feferman, eds., Model-Theoretic Logics (New York: Springer-Verlag, 1985) [1], in particular, Makowsky, J. A. Chapter XVIII: Compactness, Embeddings and Definability. 645--716, see Theorems 4.5.9, 4.6.12 and Proposition 4.6.9. For compact logics for an extended notion of model see Ziegler, M. Chapter XV: Topological Model Theory. 557--577. For logics without the relativization property it is possible to have simultaneously compactness and interpolation, while the problem is still open for logics with relativization. See Xavier Caicedo, A Simple Solution to Friedman's Fourth Problem, J. Symbolic Logic, Volume 51, Issue 3 (1986), 778-784.doi:10.2307/2274031 JSTOR 2274031 ^ Vaught, Robert L.: "Alfred Tarski's work in model theory". Journal of Symbolic Logic 51 (1986), no. 4, 869–882 ^ Robinson, A.: Non-standard analysis. North-Holland Publishing Co., Amsterdam 1966. page 48. ^ Jump up to: a b c Marker 2002, pp. 40–43. ^ Gowers, Barrow-Green & Leader 2008, pp. 639–643. ^ Jump up to: a b Terence, Tao (7 March 2009). "Infinite fields, finite fields, and the Ax-Grothendieck theorem". ^ Goldblatt, Robert (1998). Lectures on the Hyperreals. New York: Springer Verlag. pp. 10–11. ISBN 0-387-98464-X. ^ See Hodges (1993). References Boolos, George; Jeffrey, Richard; Burgess, John (2004). Computability and Logic (fourth ed.). Cambridge University Press. Chang, C.C.; Keisler, H. Jerome (1989). Model Theory (third ed.). Elsevier. ISBN 0-7204-0692-7. Dawson, John W. junior (1993). "The compactness of first-order logic: From Gödel to Lindström". History and Philosophy of Logic. 14: 15–37. doi:10.1080/01445349308837208. Hodges, Wilfrid (1993). Model theory. Cambridge University Press. ISBN 0-521-30442-3. Gowers, Timothy; Barrow-Green, June; Leader, Imre (2008). The Princeton Companion to Mathematics. Princeton: Princeton University Press. pp. 635–646. ISBN 978-1-4008-3039-8. OCLC 659590835. Marker, David (2002). Model Theory: An Introduction. Graduate Texts in Mathematics. Vol. 217. Springer. ISBN 978-0-387-98760-6. OCLC 49326991. Robinson, J. A. (1965). "A Machine-Oriented Logic Based on the Resolution Principle". Journal of the ACM. Association for Computing Machinery (ACM). 12 (1): 23–41. doi:10.1145/321250.321253. ISSN 0004-5411. S2CID 14389185. Truss, John K. (1997). Foundations of Mathematical Analysis. Oxford University Press. ISBN 0-19-853375-6. External links Compactness Theorem, Internet Encyclopedia of Philosophy. show vte Mathematical logic Categories: Mathematical logicMetatheoremsModel theoryTheorems in the foundations of mathematics