In model theory, a mathematical discipline, a β-model (from the French "bon ordre", well-ordering[1]) is a model which is correct about statements of the form "X is well-ordered".
The term was introduced by Mostowski (1959)[2][3] as a strengthening of the notion of ω-model.
In contrast to the notation for set-theoretic properties named by ordinals, such as
-indescribability, the letter β here is only denotational.
β-models appear in the study of the reverse mathematics of subsystems of second-order arithmetic.
In this context, a β-model of a subsystem of second-order arithmetic is a model M where for any Σ11 formula
243 Every β-model of second-order arithmetic is also an ω-model, since working within the model we can prove that < is a well-ordering, so < really is a well-ordering of the natural numbers of the model.
[2] There is an incompleteness theorem for β-models: if T is a recursively axiomatizable theory in the language of second-order arithmetic, analogously to how there is a model of T+"there is no model of T" if there is a model of T, there is a β-model of T+"there are no countable coded β-models of T" if there is a β-model of T. A similar theorem holds for βn-models for any natural number
[5] Axioms based on β-models provide a natural finer division of the strengths of subsystems of second-order arithmetic, and also provide a way to formulate reflection principles.
[of second-order sort], there exists a countable β-model M such that
253 (Countable ω-models are represented by their sets of integers, and their satisfaction is formalizable in the language of analysis by an inductive definition.)
Also, the theory extending KP with a canonical axiom schema for a recursively Mahlo universe (often called
)[6] is logically equivalent to the theory Δ12-CA+BI+(Every true Π13-formula is satisfied by a β-model of Δ12-CA).
proves a connection between β-models and the hyperjump: for all sets
has a hyperjump iff there exists a countable β-model
[8] A notion of β-model can be defined for models of second-order set theories (such as Morse-Kelley set theory) as a model
This logic-related article is a stub.