A complete theory is -stable, for an infinite cardinal , when for every parameter set of cardinality in a model of . Here is the type space of complete one-variable types. The theory is stable if this holds for some such ; it is omega-stable if it holds for .
For the theory of an infinite set with equality alone, a type over either names one element of or says that it differs from every member of . There are only possibilities. By contrast, a dense linear order without endpoints has the order property, producing many types corresponding to cuts and hence instability. Stability measures the number of ways a new element can relate to existing parameters, rather than simply the number of models.
A complete theory in a countable first-order language is omega-stable if over every countable set of parameters it has only countably many complete one-types. The theory of a free bijection is an example: a new point is either a specified integer translate of a parameter or lies in a new orbit. It is omega-stable but not omega-categorical.
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.