Stable Theory - Unstable Theories

Unstable Theories

Roughly speaking, a theory is unstable if one can use it to encode the ordered set of natural numbers. More precisely, if there is a model M and a formula Φ(X,Y) in 2n variables X=x₁,...,x_n and Y=y₁,...,y_n defining a relation on Mn with an infinite totally ordered subset then the theory is unstable. (Any infinite totally ordered set has a subset isomorphic to either the positive or negative integers under the usual order, so one can assume the totally ordered subset is ordered like the positive integers.) The totally ordered subset need not be definable in the theory.

The number of models of an unstable theory T of any uncountable cardinality κ≥|T| is the maximum possible number 2κ.

Examples:

• Most sufficiently complicated theories, such as set theories and Peano arithmetic, are unstable.
• The theory of the rational numbers, considered as an ordered set, is unstable. Its theory is the theory of dense linear orders without endpoints.
• The theory of addition of the natural numbers is unstable.
• Any infinite Boolean algebra is unstable.
• Any monoid with cancellation that is not a group is unstable, because if a is an element that is not a unit then the powers of a form an infinite totally ordered set under the relation of divisibility. For a similar reason any integral domain that is not a field is unstable.
• There are many unstable nilpotent groups. One example is the infinite dimensional Heisenberg group over the integers: this is generated by elements x_i, y_i, z for all natural numbers i, with the relations that any of these two generators commute except that x_i and y_i have commutator z for any i. If a_i is the element x₀x₁...x_i−1y_i then a_i and a_j have commutator z exactly when i<j, so they form an infinite total order under a definable relation, so the group is unstable.
• Real closed fields are unstable, as they are infinite and have a definable total order.