Nonforking

Nonforking is a central notion of independence in model theory, the branch of mathematical logic that studies first-order theories. It describes when a type over a larger set B does not introduce new dependencies relative to a smaller base A. In a theory T with A ⊆ B, a type p over B is said to not fork over A if it behaves as if B were built from A without adding new information about A. Forking and dividing are the combinatorial phenomena used to detect dependence.

Informally, a type p over B does not fork over A if, when realizing p over B, one can do so without creating new, A-definable constraints. Equivalently, a formula in p does not fork over A when it does not force an incompatible pattern over A. A nonforking extension is an extension of a type to a larger set that preserves this independence from A.

In stable theories nonforking satisfies standard independence properties: invariance under automorphisms, monotonicity, transitivity, extension, and symmetry. These properties allow one to reason about independence much as one reasons about dimension or linear independence in algebra. In simple theories a related notion, sometimes called Kim-independence, plays a similar role with adjusted axioms.

Examples and applications: in the theory of algebraically closed fields, nonforking corresponds to algebraic independence over subfields. Nonforking extensions then reflect adjoining elements without new algebraic relations to the base. More generally, nonforking is used to study canonical bases, dimension theory, and the structure of definable groups and fields within stable theories, as well as in related areas such as differential fields.

Origins and scope: nonforking emerged from Shelah’s stability theory as a robust notion of independence for broad classes of theories. It is a technical concept not defined in every theory, but when it exists, it provides a framework for understanding how information about a base set governs extensions and interactions among elements.