Model theory
In mathematical logic, model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models (those structures in which the statements of the theory hold).[1] The aspects investigated include the number and size of models of a theory, the relationship of different models to each other, and their interaction with the formal language itself. In particular, model theorists also investigate the sets that can be defined in a model of a theory, and the relationship of such definable sets to each other. As a separate discipline, model theory goes back to Alfred Tarski, who first used the term "Theory of Models" in publication in 1954.[2] Since the 1970s, the subject has been shaped decisively by Saharon Shelah's stability theory.
This article is about the mathematical discipline. For the informal notion in other parts of mathematics and science, see Mathematical model.
Compared to other areas of mathematical logic such as proof theory, model theory is often less concerned with formal rigour and closer in spirit to classical mathematics.
This has prompted the comment that "if proof theory is about the sacred, then model theory is about the profane".[3]
The applications of model theory to algebraic and Diophantine geometry reflect this proximity to classical mathematics, as they often involve an integration of algebraic and model-theoretic results and techniques. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.
The most prominent scholarly organization in the field of model theory is the Association for Symbolic Logic.
Definability[edit]
Definable sets[edit]
In model theory, definable sets are important objects of study. For instance, in the formula
Non-elementary model theory[edit]
Model-theoretic results have been generalised beyond elementary classes, that is, classes axiomatisable by a first-order theory.
Model theory in higher-order logics or infinitary logics is hampered by the fact that completeness and compactness do not in general hold for these logics. This is made concrete by Lindstrom's theorem, stating roughly that first-order logic is essentially the strongest logic in which both the Löwenheim-Skolem theorems and compactness hold. However, model theoretic techniques have been developed extensively for these logics too.[38] It turns out, however, that much of the model theory of more expressive logical languages is independent of Zermelo-Fraenkel set theory.[39]
More recently, alongside the shift in focus to complete stable and categorical theories, there has been work on classes of models defined semantically rather than axiomatised by a logical theory.
One example is homogeneous model theory, which studies the class of substructures of arbitrarily large homogeneous models. Fundamental results of stability theory and geometric stability theory generalise to this setting.[40]
As a generalisation of strongly minimal theories, quasiminimally excellent classes are those in which every definable set is either countable or co-countable. They are key to the model theory of the complex exponential function.[41]
The most general semantic framework in which stability is studied are abstract elementary classes, which are defined by a strong substructure relation generalising that of an elementary substructure. Even though its definition is purely semantic, every abstract elementary class can be presented as the models of a first-order theory which omit certain types. Generalising stability-theoretic notions to abstract elementary classes is an ongoing research program.[42]
Selected applications[edit]
Among the early successes of model theory are Tarski's proofs of quantifier elimination for various algebraically interesting classes, such as the real closed fields, Boolean algebras and algebraically closed fields of a given characteristic. Quantifier elimination allowed Tarski to show that the first-order theories of real-closed and algebraically closed fields as well as the first-order theory of Boolean algebras are decidable, classify the Boolean algebras up to elementary equivalence and show that the theories of real-closed fields and algebraically closed fields of a given characteristic are unique. Furthermore, quantifier elimination provided a precise description of definable relations on algebraically closed fields as algebraic varieties and of the definable relations on real-closed fields as semialgebraic sets [43][44]
In the 1960s, the introduction of the ultraproduct construction led to new applications in algebra. This includes Ax's work on pseudofinite fields, proving that the theory of finite fields is decidable,[45] and Ax and Kochen's proof of as special case of Artin's conjecture on diophantine equations, the Ax-Kochen theorem.[46] The ultraproduct construction also led to Abraham Robinson's development of nonstandard analysis, which aims to provide a rigorous calculus of infinitesimals.[47]
More recently, the connection between stability and the geometry of definable sets led to several applications from algebraic and diophantine geometry, including Ehud Hrushovski's 1996 proof of the geometric Mordell-Lang conjecture in all characteristics[48] In 2001, similar methods were used to prove a generalisation of the Manin-Mumford conjecture.
In 2011, Jonathan Pila applied techniques around o-minimality to prove the André-Oort conjecture for products of Modular curves.[49]
In a separate strand of inquiries that also grew around stable theories, Laskowski showed in 1992 that NIP theories describe exactly those definable classes that are PAC-learnable in machine learning theory. This has led to several interactions between these separate areas. In 2018, the correspondence was extended as Hunter and Chase showed that stable theories correspond to online learnable classes.[50]
History[edit]
Model theory as a subject has existed since approximately the middle of the 20th century, and the name was coined by Alfred Tarski, a member of the Lwów–Warsaw school, in 1954.[51] However some earlier research, especially in mathematical logic, is often regarded as being of a model-theoretical nature in retrospect.[52] The first significant result in what is now model theory was a special case of the downward Löwenheim–Skolem theorem, published by Leopold Löwenheim in 1915. The compactness theorem was implicit in work by Thoralf Skolem,[53] but it was first published in 1930, as a lemma in Kurt Gödel's proof of his completeness theorem. The Löwenheim–Skolem theorem and the compactness theorem received their respective general forms in 1936 and 1941 from Anatoly Maltsev.
The development of model theory as an independent discipline was brought on by Alfred Tarski during the interbellum. Tarski's work included logical consequence, deductive systems, the algebra of logic, the theory of definability, and the semantic definition of truth, among other topics. His semantic methods culminated in the model theory he and a number of his Berkeley students developed in the 1950s and '60s.
In the further history of the discipline, different strands began to emerge, and the focus of the subject shifted. In the 1960s, techniques around ultraproducts became a popular tool in model theory.[54] At the same time, researchers such as James Ax were investigating the first-order model theory of various algebraic classes, and others such as H. Jerome Keisler were extending the concepts and results of first-order model theory to other logical systems. Then, inspired by Morley's problem, Shelah developed stability theory. His work around stability changed the complexion of model theory, giving rise to a whole new class of concepts. This is known as the paradigm shift. [55]
Over the next decades, it became clear that the resulting stability hierarchy is closely connected to the geometry of sets that are definable in those models; this gave rise to the subdiscipline now known as geometric stability theory. An example of an influential proof from geometric model theory is Hrushovski's proof of the Mordell–Lang conjecture for function fields.[56]