# 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.