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.
|[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=14510