In model theory

Model theory

In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....

, an abstract elementary class, or AEC for short, is a class of models with a partial order similar to the relation of a substructure

Substructure

In mathematical logic, an substructure or subalgebra is a structure whose domain is a subset of that of a bigger structure, and whose functions and relations are the traces of the functions and relations of the bigger structure...

 of an elementary class

Elementary class

In the branch of mathematical logic called model theory, an elementary class is a class consisting of all structures satisfying a fixed first-order theory.- Definition :...

 in first-order

First-order logic

First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...

 model theory. They were studied by .

Definition

, for K a class of structures in some language L, is an AEC if it has the following properties:

• is a partial order on K
• If then
• Isomorphisms: K is closed under isomorphisms
  Isomorphism
  In abstract algebra, an isomorphism is a mapping between objects that shows a relationship between two properties or operations.  If there exists an isomorphism between two structures, the two structures are said to be isomorphic.  In a certain sense, isomorphic structures are...
  
  , and if such that , , and then
• Coherence: If , and then
• Tarski-Vaught Chain
  Total order
  In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
  
   Axioms If is a chain, ie.
  • 
  • if , for all , then
• Löwenheim–Skolem
  Löwenheim–Skolem theorem
  In mathematical logic, the Löwenheim–Skolem theorem, named for Leopold Löwenheim and Thoralf Skolem, states that if a countable first-order theory has an infinite model, then for every infinite cardinal number κ it has a model of size κ...
  
   Axiom There exists a cardinal
  Cardinal number
  In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
  
   denoted by , such that if A is a subset of the universe of M, then there is M' in K whose universe contains A such that and

Examples

• Elementary classes
  Elementary class
  In the branch of mathematical logic called model theory, an elementary class is a class consisting of all structures satisfying a fixed first-order theory.- Definition :...
  
   are the most basic example of an AEC, when taken with taken to be elementary substructure
  Elementary substructure
  In model theory, a field within mathematical logic, two structures M and N of the same signature σ are called elementarily equivalent if they satisfy the same first-order σ-sentences....
  
  .
• If is a cardinal, is a theory in the infinitary logic
  Infinitary logic
  An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. Some infinitary logics may have different properties from those of standard first-order logic. In particular, infinitary logics may fail to be compact or complete. Notions of compactness and...
  
   , and is a fragment of the containing , then is an AEC.

Shelah's Categoricity Conjecture

The work done on AECs is in large part done to prove Shelah's categoricity conjecture, which is a conjecture

Conjecture

A conjecture is a proposition that is unproven but is thought to be true and has not been disproven. Karl Popper pioneered the use of the term "conjecture" in scientific philosophy. Conjecture is contrasted by hypothesis , which is a testable statement based on accepted grounds...

 analogous to Morley's categoricity theorem

Morley's categoricity theorem

In model theory, a branch of mathematical logic, a theory is κ-categorical if it has exactly one model of cardinality κ up to isomorphism....

 in first-order model theory.

The conjecture states, in simple terms, for every AEC K there is some cardinal (which depends only on ) such that if K is categorical at some , ie. every model of size is isomorphic, then K is categorical at all .

Results

The following are some some important results about AECs:

• Shelah's Presentation Theorem: Any AEC can be written as a PC
  Pseudoelementary class
  In logic, a pseudoelementary class is a class of structures derived from an elementary class by omitting some of its sorts and relations. It is the mathematical logic counterpart of the notion in category theory of a forgetful functor, and in physics of hidden variable theories purporting to...
  
   class, where the language is of size and we omit at most many types.