Beta-model

Short description: A class of "well-behaved" models in set theory

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 [math]\displaystyle{ \xi }[/math]-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 [math]\displaystyle{ \phi }[/math] with parameters from M, [math]\displaystyle{ (\omega,M,+,\times,0,1,\lt )\vDash\phi }[/math] iff [math]\displaystyle{ (\omega,\mathcal P(\omega),+,\times,0,1,\lt )\vDash\phi }[/math].^p. 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.

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 β_n-models for any natural number [math]\displaystyle{ n\geq 1 }[/math].

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 [math]\displaystyle{ \mathsf{ATR}_0 }[/math], [math]\displaystyle{ \Pi^1_1\mathsf{-CA}_0 }[/math] is equivalent to the statement "for all [math]\displaystyle{ X }[/math] [of second-order sort], there exists a countable β-model M such that [math]\displaystyle{ X\in M }[/math].^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 [math]\displaystyle{ KPM }[/math]) is logically equivalent to the theory Δ12-CA+BI+(Every true Π13-formula is satisfied by a β-model of Δ12-CA).

Additionally, [math]\displaystyle{ \mathsf{ACA}_0 }[/math] proves a connection between β-models and the hyperjump: for all sets [math]\displaystyle{ X }[/math] of integers, [math]\displaystyle{ X }[/math] has a hyperjump iff there exists a countable β-model [math]\displaystyle{ M }[/math] such that [math]\displaystyle{ X\in M }[/math].^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 [math]\displaystyle{ (M, \mathcal X) }[/math] such that the membership relations of [math]\displaystyle{ (M, \mathcal X) }[/math] is well-founded, and for any relation [math]\displaystyle{ R\in\mathcal X }[/math], [math]\displaystyle{ (M, \mathcal X)\vDash }[/math]"[math]\displaystyle{ R }[/math] is well-founded" iff [math]\displaystyle{ R }[/math] 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}