Beta-model

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 ${\displaystyle \phi }$ with parameters from M, ${\displaystyle (\omega ,M,+,\times ,0,1,<)\vDash \phi }$ iff ${\displaystyle (\omega ,{\mathcal {P}}(\omega ),+,\times ,0,1,<)\vDash \phi }$.^[4]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.^[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 β_n-models^{[clarification needed]} for any natural number ${\displaystyle n\geq 1}$.^[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. For example, over ${\displaystyle {\mathsf {ATR}}_{0}}$, ${\displaystyle \Pi _{1}^{1}{\mathsf {-CA}}_{0}}$ is equivalent to the statement "for all ${\displaystyle X}$ [of second-order sort], there exists a countable β-model M such that ${\displaystyle X\in M}$.^[4]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 ${\displaystyle KPM}$)^[6] is logically equivalent to the theory Δ¹
₂-CA+BI+(Every true Π¹
₃-formula is satisfied by a β-model of Δ¹
₂-CA).^[7]

Additionally, ${\displaystyle {\mathsf {ACA}}_{0}}$ proves a connection between β-models and the hyperjump: for all sets ${\displaystyle X}$ of integers, ${\displaystyle X}$ has a hyperjump iff there exists a countable β-model ${\displaystyle M}$ such that ${\displaystyle X\in M}$.^[4]p. 251

Every β-model of comprehension is elementarily equivalent to an ω-model which is not a 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 ${\displaystyle (M,{\mathcal {X}})}$ such that the membership relations of ${\displaystyle (M,{\mathcal {X}})}$ is well-founded, and for any relation ${\displaystyle R\in {\mathcal {X}}}$, ${\displaystyle (M,{\mathcal {X}})\vDash }$"${\displaystyle R}$ is well-founded" iff ${\displaystyle R}$ is in fact well-founded. While there is no least transitive model of MK, there is a least β-model of MK.^{[9]pp. 17,154–156}