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)