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, " is finite for  -categorical  " Trans. Amer. Math. Soc. , 181 (1973) pp. 37–51 |
| [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