A complete theory is -stable when for every parameter set of cardinality . It is omega-stable when this holds for .
For a complete theory, the following are equivalent: the theory is stable; no formula has the order property; and every complete type over every model is definable.
A type over a model is definable when, for every formula , there is an -formula such thatfor every tuple from .
If is stable, , and is definable with parameters from , then is definable with parameters from . Apply definability of the type over of the parameter defining .
For a countable theory, omega-stability implies for every parameter set . Total transcendence gives each type a finite base and a definition over it; counting finite bases and formulas proves the bound.
A formula has the order property for when arbitrarily long tuples exist with exactly when . Compactness realizes this pattern along every linear order.
If a theory has the order property, a linear order with more than cuts over a dense subset of size at most produces more than distinct types over that subset. Hence a theory with the order property is not -stable for any .
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Stable theory is a branch of model theory, which is a field of mathematical logic. Introduced by Morley in the early 1960s, stable theory primarily concerns the study of structures that satisfy certain stability conditions. Stability, here, refers to a way of categorizing theories based on their behavior in terms of definability and the complexity of their types. A theory is said to be stable if its behavior can be well-controlled, especially in terms of the number of types over various sets.