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 $ M $ of a complete first-order theory $ T $. It is defined inductively by: $ { \mathop{\rm Morleyrank} } \theta ( x ) \geq \alpha + 1 $ if there is an elementary extension $ N $ of $ M $ and infinitely many formulas $ \phi _ {i} ( x ) $( $ i < \omega $) with parameters from $ N $ such that the $ \phi _ {i} ( x ) $ are pairwise inconsistent, $ N \vDash \phi _ {i} ( x ) \rightarrow \theta ( x ) $ for all $ i $ and $ { \mathop{\rm Morleyrank} } \phi _ {i} ( x ) \geq \alpha $ for all $ i $. For $ \delta $ a limit ordinal (cf. also Ordinal number), $ { \mathop{\rm Morleyrank} } \theta ( x ) \geq \delta $ if $ { \mathop{\rm Morleyrank} } \theta ( x ) \geq \alpha $ for all $ \alpha < \delta $. The Morley rank of $ \theta ( x ) $ is said to be equal to $ \alpha $ if it is greater than or equal to $ \alpha $ but not greater than or equal to $ \alpha + 1 $. The Morley rank of $ \theta ( x ) $ is said to be $ \infty $( 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 $ T $ such that for some uncountable cardinal number $ \kappa $, $ T $ has a unique model of cardinality $ \kappa $. Morley showed that a theory $ T $ satisfying the latter condition has a unique model of cardinality $ \lambda $ for any uncountable cardinal $ \lambda $. 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 $ ACF _ {0} $ of algebraically closed fields of characteristic $ 0 $. The field $ \mathbf C $ of complex numbers is a model, and for a formula $ \phi ( x _ {1} \dots x _ {n} ) $ with parameters in $ \mathbf C $ defining a non-singular algebraic variety $ V $, the Morley rank of $ \phi $ is precisely the dimension of $ V $ 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)