Morley rank
An important notion and tool in model theory, a branch of mathematical logic. The Morley rank is an ordinal-valued dimension associated to first-order formulas with parameters from a model of a complete first-order theory
. It is defined inductively by:
if there is an elementary extension
of
and infinitely many formulas
(
) with parameters from
such that the
are pairwise inconsistent,
for all
and
for all
. For
a limit ordinal (cf. also Ordinal number),
if
for all
. The Morley rank of
is said to be equal to
if it is greater than or equal to
but not greater than or equal to
. The Morley rank of
is said to be
(or undefined) if it is not equal to any ordinal.
The Morley rank was introduced by M. Morley [a2] in his study of countable theories such that for some uncountable cardinal number
,
has a unique model of cardinality
. Morley showed that a theory
satisfying the latter condition has a unique model of cardinality
for any uncountable cardinal
. An important part of his work was to show that every formula has ordinal-valued Morley rank. Subsequently, J.T. Baldwin [a1] showed that under Morley's hypothesis, every formula has finite Morley rank.
A classical example of Morley rank occurs in the (complete) theory of algebraically closed fields of characteristic
. The field
of complex numbers is a model, and for a formula
with parameters in
defining a non-singular algebraic variety
, the Morley rank of
is precisely the dimension of
as a complex manifold.
Following the example of Morley rank, S. Shelah [a3] defined a host of rank-functions associated to formulas in first-order theories, which play an important role in classification theory.
See also Group of finite Morley rank.
References
[a1] | J.T. Baldwin, "![]() ![]() ![]() |
[a2] | M.D. Morley, "Categoricity in power" Trans. Amer. Math. Soc. , 114 (1965) pp. 514–538 |
[a3] | S. Shelah, "Classification theory and the number of non-isomorphic models" , North-Holland (1990) (Edition: Revised) |
Morley rank. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Morley_rank&oldid=47900