Forking

(in logic)

A notion introduced by S. Shelah [a8]. The general theory of forking is also known as stability theory, but more commonly, non-forking (the negation of forking) is defined as a certain well-behaved relation between a type and its extension (cf. Types, theory of).

Let $ M $ be a sufficiently saturated model of a theory $ T $ in a countable first-order language (cf. also Formal language; Model (in logic); Model theory). Given an $ n $- tuple of variables $ {\overline{x}\; } $ and $ A \subset M $, a collection of formulas $ \phi ( {\overline{x}\; } , {\overline{a}\; } ) $ with parameters $ {\overline{a}\; } $ in $ A $ is called an $ n $- type over $ A $. For simplicity, only $ 1 $- types will be considered; these are simply called types over $ A $. A complete type is one which is maximal consistent. Let $ S ( A ) $ be the set of complete types over $ A $.

Given a type $ t = t ( x ) $ and a formula $ \phi = \phi ( x, {\overline{y}\; } ) $, one defines the Morley $ \phi $- rank of $ t $, $ \phi roman \AAh { \mathop{\rm rk} } ( t ) $, inductively as follows: $ \phi roman \AAh { \mathop{\rm rk} } ( t ) \geq 0 $ if $ t $ is consistent, for each natural number $ n $, $ \phi roman \AAh { \mathop{\rm rk} } ( t ) \geq n + 1 $ if for every finite $ s \subset t $ and natural number $ m $ there are collections $ p _ {1} \dots p _ {m} $ of $ \phi $- formulas (with parameters from $ M $) such that:

i) for $ i \neq j $, $ p _ {i} $ and $ p _ {j} $ are contradictory, i.e. for some $ {\overline{a}\; } $, $ \phi ( x, {\overline{a}\; } ) $ belongs to one of $ p _ {i} $ and $ p _ {j} $, and $ \neg \phi ( x, {\overline{a}\; } ) $ belongs to the other;

ii) $ \phi roman \AAh { \mathop{\rm rk} } ( s \cup p _ {i} ) \geq n $.

Assume that $ T $ is stable, i.e. for some infinite $ \kappa $, whenever $ | A | \leq \kappa $, then also $ | {S ( A ) } | \leq \kappa $. (Equivalently, $ \phi roman \AAh { \mathop{\rm rk} } ( t ) < \infty $ for every type $ t $ and formula $ \phi $.) Let $ A \subset B $, $ t \in S ( A ) $, $ u \in S ( B ) $ be such that $ u \supset t $. Then $ u $ is called a non-forking extension of $ t $, or it is said that $ u $ does not fork over $ A $, if for every formula $ \phi $ with $ \phi ( x, {\overline{b}\; } ) \in u $,

$$ \phi roman \AAh { \mathop{\rm rk} } ( p ) = \phi roman \AAh { \mathop{\rm rk} } ( p \cap \phi ( x, {\overline{b}\; } ) ) , $$

where $ p \cap \phi ( x, {\overline{b}\; } ) $ denotes the set $ \{ {\theta \wedge \phi ( x, {\overline{b}\; } ) } : {\theta \in p } \} $.

Let $ t \Sbs u $ mean that $ u $ is a non-forking extension of $ t $. Then $ \Sbs $ is the unique relation on complete types satisfying the following Lascar axioms:

1) $ \Sbs $ is preserved under automorphisms of $ M $;

2) if $ t \subset u \subset v $, then $ t \Sbs v $ if and only if $ t \Sbs u $ and $ u \Sbs v $;

3) for any $ t \in S ( A ) $ and $ B \supset A $ there exists a $ u \in S ( B ) $ such that $ t \Sbs u $;

4) for any $ t \in S ( A ) $ there exist countable $ A _ {0} \subset A $ and $ t _ {0} \Sbs t $, where $ t _ {0} $ is the restriction of $ t $ to formulas with parameters from $ A _ {0} $;

5) for any $ t \in S ( A ) $ and $ A \subset B $,

$$ \left | {\left \{ {u \in S ( B ) } : {t \Sbs u } \right \} } \right | \leq 2 ^ {\aleph _ {0} } . $$

The ultrapower construction (cf. also Ultrafilter) gives a systematic way of building non-forking extensions [a4].

For $ c \in M $ one writes $ { \mathop{\rm tp} } ( c/A ) $ for the type in $ S ( A ) $ realized by $ c $. Given a set $ A $ and $ b,c \in M $, the following important symmetry property holds: $ { \mathop{\rm tp} } ( b/A \cup \{ c \} ) $ does not fork over $ A $ if and only if $ { \mathop{\rm tp} } ( c/A \cup \{ b \} ) $ does not fork over $ A $. If either holds, one says that $ b $, $ c $ are independent over $ A $, and this notion is viewed as a generalization of algebraic independence.

Given $ t \in S ( A ) $, $ B \supset A $, $ u \in S ( B ) $, and $ u \supset t $, one says that $ u $ is an heir of $ t $ if for every $ \phi ( x, {\overline{y}\; } ) $( with parameters in $ A $), $ \phi ( x, {\overline{b}\; } ) \in u $ for some $ {\overline{b}\; } $ in $ B $ if and only if $ \phi ( x, {\overline{a}\; } ) \in t $ for some $ {\overline{a}\; } $ in $ A $. One says that $ u $ is definable over $ A $ if for every $ \phi ( x, {\overline{y}\; } ) $ there is a formula $ \theta ( {\overline{y}\; } ) $ with parameters from $ A $ such that for any $ {\overline{b}\; } $ in $ B $, $ \phi ( x, {\overline{b}\; } ) \in u $ if and only if $ M \vDash \theta ( {\overline{b}\; } ) $.

$ u $ is said to be a coheir of $ t $ if $ u $ is finitely satisfiable in $ A $. So, for $ b,c \in M $, $ { \mathop{\rm tp} } ( c/ A \cup \{ b \} ) $ is an heir of $ { \mathop{\rm tp} } ( c/A ) $ if and only if $ { \mathop{\rm tp} } ( b/A \cup \{ c \} ) $ is a coheir of $ { \mathop{\rm tp} } ( b/A ) $.

If $ A $ is an elementary submodel of $ M $, then $ u \Sps t $ if and only if $ u $ is an heir of $ t $ if and only if $ u $ is definable over $ A $. In particular, in that case $ t $ has a unique non-forking extension over any $ B \supset A $. Then it follows from the forking symmetry that when $ A $ is an elementary submodel, $ { \mathop{\rm tp} } ( b/A \cup \{ c \} ) $ being a coheir of $ { \mathop{\rm tp} } ( b/A ) $ is equivalent to being an heir.

For a comprehensive introduction of forking see [a1], [a2], [a4], [a5], and [a9]. For applications in algebra, see [a7] and [a6].

The techniques of forking have been extended to unstable theories. In [a2], this is done by considering only types that satisfy stable conditions. In [a3], types are viewed as probability measures and forking is treated as a special kind of measure extension. The stability assumption is then weakened to theories that do not have the independence property.

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