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
,
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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 |
Forking. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Forking&oldid=46953