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

Given a type and a formula , one defines the Morley -rank of , , inductively as follows: if is consistent, for each natural number , if for every finite and natural number there are collections of -formulas (with parameters from ) such that:

i) for , and are contradictory, i.e. for some , belongs to one of and , and belongs to the other;

ii) .

Assume that is stable, i.e. for some infinite , whenever , then also . (Equivalently, for every type and formula .) Let , , be such that . Then is called a non-forking extension of , or it is said that does not fork over , if for every formula with ,

where denotes the set .

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

1) is preserved under automorphisms of ;

2) if , then if and only if and ;

3) for any and there exists a such that ;

4) for any there exist countable and , where is the restriction of to formulas with parameters from ;

5) for any and ,

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

For one writes for the type in realized by . Given a set and , the following important symmetry property holds: does not fork over if and only if does not fork over . If either holds, one says that , are independent over , and this notion is viewed as a generalization of algebraic independence.

Given , , , and , one says that is an heir of if for every (with parameters in ), for some in if and only if for some in . One says that is definable over if for every there is a formula with parameters from such that for any in , if and only if .

is said to be a coheir of if is finitely satisfiable in . So, for , is an heir of if and only if is a coheir of .

If is an elementary submodel of , then if and only if is an heir of if and only if is definable over . In particular, in that case has a unique non-forking extension over any . Then it follows from the forking symmetry that when is an elementary submodel, being a coheir of 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.

References

[a1]	J.T. Baldwin, "Fundamentals of stability theory" , Springer (1987)
[a2]	V. Harnik, L. Harrington, "Fundamentals of forking" Ann. Pure and Applied Logic , 26 (1984) pp. 245–286
[a3]	H.J. Keisler, "Measures and forking" Ann. Pure and Applied Logic , 34 (1987) pp. 119–169
[a4]	D. Lascar, B. Poizat, "An introduction to forking" J. Symb. Logic , 44 (1979) pp. 330–350
[a5]	A. Pillay, "Introduction to stability theory" , Oxford Univ. Press (1983)
[a6]	A. Pillay, "The geometry of forking and groups of finite Morley rank" J. Symb. Logic , 60 (1995) pp. 1251–1259
[a7]	M. Prest, "Model theory and modules" , Cambridge Univ. Press (1988)
[a8]	S. Shelah, "Classification theory and the number of non-isomorphic models" , North-Holland (1990) (Edition: Revised)
[a9]	M. Makkai, "A survey of basic stability theory" Israel J. Math. , 49 (1984) pp. 181–238