Beta-model

In model theory, a mathematical discipline, a β-model (from the French "bon ordre", well-ordering ) is a model which is correct about statements of the form "X is well-ordered". The term was introduced by Mostowski (1959) as a strengthening of the notion of ω-model. In contrast to the notation for set-theoretic properties named by ordinals, such as $\xi$-indescribability, the letter β here is only denotational.

In analysis
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 $$\phi$$ with parameters from M, $$(\omega,M,+,\times,0,1,<)\vDash\phi$$ iff $$(\omega,\mathcal P(\omega),+,\times,0,1,<)\vDash\phi$$. p. 243 Every β-model of second-order arithmetic is also an ω-model, since working within the model we can prove that &lt; is a well-ordering, so &lt; really is a well-ordering of the natural numbers of the model.

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 $$n\geq 1$$.

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. For example, over $$\mathsf{ATR}_0$$, $$\Pi^1_1\mathsf{-CA}_0$$ is equivalent to the statement "for all $$X$$ [of second-order sort], there exists a countable β-model M such that $$X\in M$$. p. 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 $$KPM$$) is logically equivalent to the theory Δ$1 2$-CA+BI+(Every true Π$1 3$-formula is satisfied by a β-model of Δ$1 2$-CA).

Additionally, $$\mathsf{ACA}_0$$ proves a connection between β-models and the hyperjump: for all sets $$X$$ of integers, $$X$$ has a hyperjump iff there exists a countable β-model $$M$$ such that $$X\in M$$. p. 251

In set theory
A notion of β-model can be defined for models of second-order set theories (such as Morse-Kelley set theory) as a model $$(M, \mathcal X)$$ such that the membership relations of $$(M, \mathcal X)$$ is well-founded, and for any relation $$R\in\mathcal X$$, $$(M, \mathcal X)\vDash$$"$$R$$ is well-founded" iff $$R$$ is in fact well-founded. While there is no least transitive model of MK, there is a least β-model of MK. pp.17,154--156