Model Theory | Research & Encyclopedia Articles

Model Theory

Model theory is a branch of mathematical logic concerned with the study of formal theories viewed as mathematical structures or objects. Those mathematical structures are studied by examining the first order sentences that describe those structures and the sets in those structures that are defined by first order formulas. Model theory is that part of mathematics that demonstrates how to apply mathematical logic to the study of structures in mathematics. Those structures investigated in model theory can be expressed in a formal language that usually consists of first order statements.

Model theory's fundamental tools are interpretations. Mathematical truth is like all truth in that it is relative to specific situations and depends upon how and where it is interpreted. This is because of the language used to express mathematical ideas rather than to mathematics itself. The theory itself is generally one consisting of interpretations of axiomatic set theory. Axiomatic set theory is a specific version of set theory, a branch of mathematics closely associated to logic, consisting of axioms that are taken as un-interpreted as opposed to being formalizations of pre-existing truths. Mathematical structures that obey axioms in a system are considered the models of that system. The second order axioms of analysis are known to have real numbers as their model. Nonstandard analysis is a form of model theory in which the axioms are weakened to include only the first order axioms. Model theory itself utilizes the full power of set theory.

Model theory also involves investigations concerned with the expressive power of formal languages in the sense of what they can say about specific mathematical structures. Model theory is the ultimate abstraction in the sense of its methods of analysis yet it has immediate applications to practical mathematics. An alternative to model theory is the view that deduction depends on formal rules of inference like the rules of logical calculus. The differences between model theory and deduction are similar to the differences in logic between proof-theoretic methods that are based on truth tables and proof-theoretic methods that are based on formal rules. The two main functions model theory attempts to perform are explaining the relationship between language and experience and specifying the idea of logical consequence. Classical model theory can be thought of as the act of dealing with static relationships among individuals.

The origins of model theory can be traced back to the continuum hypothesis proposed by Cantor in 1878. In this hypothesis he put forth the idea that every infinite subset of the continuum is either countable or has the cardinality of the continuum. This hypothesis was part of logic and more specifically a part of set theory. Hilbert understood the importance of this hypothesis and in the early 1900s sought to propose methods to more fully understand and prove the hypothesis. In 1902 Zermelo adopted Hilbert's ideas and published his first work on set theory. In 1905 he began to attempt to axiomatic set theory and in 1908 published his axiomatic system although he failed to prove consistency. In 1922 Skolem and Fraenkel, working independently, proceeded to improve Zermelo's axiom system and in doing so created the most commonly used system for axiomatic set theory to date. Later, Skolem extended Löwenheim's work from 1915 and formulated a nonstandard model of arithmetic in 1919. This was the first introduction of model theory but in a weaker form. Malcev's first publications in about 1933 were on model theory and these ideas later appeared in Robinson's pioneering work on model theory, for which he received his Ph.D. in 1949, and nonstandard analysis, which he introduced in 1961. Tarski continued the advancement of model theory by developing a semantic method to more thoroughly study formal scientific languages. After attending a course taught by Tarski in 1946, Lyndon began work on model theory and eventually led to the development of group theory. Mostowski also began working on the formulation and understanding of model theory at about the same time under the direction of Tarski. In 1975 a collection of works were published called Model theory and algebra as a memorial tribute to Robinson who had died a year earlier. Robinson made significant advances in model theory and is attributed with inventing nonstandard analysis, a weakened form of model theory. More recently there has been a fruitful interplay between model theory and algebra. From this there have been methods developed in a pure model-theoretic setting that have been applied in such areas as group theory, number theory, and algebraic geometry. Model theory has also enjoyed an extensive interaction with the field of infinite permutation groups. Research continues that not only employs model theory but contributes to its advancement as well.