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An important notion and tool in [[Model theory|model theory]], a branch of [[Mathematical logic|mathematical logic]]. The Morley rank is an ordinal-valued dimension associated to first-order formulas with parameters from a model <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m1102001.png" /> of a complete first-order theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m1102002.png" />. It is defined inductively by: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m1102003.png" /> if there is an elementary extension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m1102004.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m1102005.png" /> and infinitely many formulas <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m1102006.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m1102007.png" />) with parameters from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m1102008.png" /> such that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m1102009.png" /> are pairwise inconsistent, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020010.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020012.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020013.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020014.png" /> a limit ordinal (cf. also [[Ordinal number|Ordinal number]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020015.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020016.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020017.png" />. The Morley rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020018.png" /> is said to be equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020019.png" /> if it is greater than or equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020020.png" /> but not greater than or equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020021.png" />. The Morley rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020022.png" /> is said to be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020023.png" /> (or undefined) if it is not equal to any ordinal.
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The Morley rank was introduced by M. Morley [[#References|[a2]]] in his study of countable theories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020024.png" /> such that for some uncountable [[Cardinal number|cardinal number]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020026.png" /> has a unique model of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020027.png" />. Morley showed that a theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020028.png" /> satisfying the latter condition has a unique model of cardinality <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020029.png" /> for any uncountable cardinal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020030.png" />. An important part of his work was to show that every formula has ordinal-valued Morley rank. Subsequently, J.T. Baldwin [[#References|[a1]]] showed that under Morley's hypothesis, every formula has finite Morley rank.
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A classical example of Morley rank occurs in the (complete) theory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020032.png" /> of algebraically closed fields of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020033.png" />. The field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020034.png" /> of complex numbers is a model, and for a formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020035.png" /> with parameters in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020036.png" /> defining a non-singular [[Algebraic variety|algebraic variety]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020037.png" />, the Morley rank of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020038.png" /> is precisely the [[Dimension|dimension]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m110/m110200/m11020039.png" /> as a [[Complex manifold|complex manifold]].
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An important notion and tool in [[Model theory|model theory]], a branch of [[Mathematical logic|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|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 [[#References|[a2]]] in his study of countable theories  $  T $
 +
such that for some uncountable [[Cardinal number|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 [[#References|[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|algebraic variety]] $  V $,  
 +
the Morley rank of $  \phi $
 +
is precisely the [[Dimension|dimension]] of $  V $
 +
as a [[Complex manifold|complex manifold]].
  
 
Following the example of Morley rank, S. Shelah [[#References|[a3]]] defined a host of rank-functions associated to formulas in first-order theories, which play an important role in classification theory.
 
Following the example of Morley rank, S. Shelah [[#References|[a3]]] defined a host of rank-functions associated to formulas in first-order theories, which play an important role in classification theory.

Latest revision as of 08:01, 6 June 2020


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)
How to Cite This Entry:
Morley rank. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Morley_rank&oldid=14510
This article was adapted from an original article by A. Pillay (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article