, viewed as a structure with binary functions for addition and multiplication and constants for 0 and 1 of the natural numbers, for example, an element
This generalises the analogous concepts from algebra; for instance, a subgroup is a substructure in the signature with multiplication and inverse.
Every embedding is an injective homomorphism, but the converse holds only if the signature contains no relation symbols, such as in groups or fields.
A field or a vector space can be regarded as a (commutative) group by simply ignoring some of its structure.
For example, the theory of algebraically closed fields in the signature σring = (×,+,−,0,1) has quantifier elimination.
[11] This means that in an algebraically closed field, every formula is equivalent to a Boolean combination of equations between polynomials.
In this way, one can study definable groups and fields in general structures, for instance, which has been important in geometric stability theory.
, a (partial) n-type over A is a set of formulas p with at most n free variables that are realised in an elementary extension
However, a compactness argument shows that there is an elementary extension of the real number line in which there is an element larger than any integer.
By Stone's representation theorem for Boolean algebras there is a natural dual topological space, which consists exactly of the complete
Since two models of different sizes cannot possibly be isomorphic, only finite structures can be described by a categorical theory.
It turns out that the question of κ-categoricity depends critically on whether κ is bigger than the cardinality of the language (i.e.
This means that every countable dense linear order is order-isomorphic to the rational number line.
Michael Morley showed in 1963 that there is only one notion of uncountable categoricity for theories in countable languages.
A key factor in the structure of the class of models of a first-order theory is its place in the stability hierarchy.
There are also analogues of Morley rank which are well-defined if and only if a theory is superstable (U-rank) or merely stable (Shelah's
[38] As a generalisation of strongly minimal theories, quasiminimally excellent classes are those in which every definable set is either countable or co-countable.
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.
[44] The ultraproduct construction also led to Abraham Robinson's development of nonstandard analysis, which aims to provide a rigorous calculus of infinitesimals.
[45] 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[46] 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.
[48] 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.
[50] 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.
In the further history of the discipline, different strands began to emerge, and the focus of the subject shifted.
His work around stability changed the complexion of model theory, giving rise to a whole new class of concepts.
[58] The model-theoretic viewpoint has been useful in set theory; for example in Kurt Gödel's work on the constructible universe, which, along with the method of forcing developed by Paul Cohen can be shown to prove the (again philosophically interesting) independence of the axiom of choice and the continuum hypothesis from the other axioms of set theory.
For example, if the Continuum Hypothesis holds then every countable model has an ultrapower which is saturated (in its own cardinality).
Similarly, if the Generalized Continuum Hypothesis holds then every model has a saturated elementary extension.
Finally, some questions arising from model theory (such as compactness for infinitary logics) have been shown to be equivalent to large cardinal axioms.