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)K. R. Apt, W. Marek,
Second-order Arithmetic and Some Related Topics
(1973), p. 181 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 set theory

It's a consequence of Shoenfield's absoluteness theorem that the constructible universe L is a β-model.

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 φ with parameters from M, (ω,M,+,×,0,1,<)⊨φ iff (ω,P(ω),+,×,0,1,<)⊨φ.S. G. Simpson, ''Subsystems of Second-Order Arithmetic (2009)^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.

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 ATR₀, Π-CA₀ is equivalent to the statement "for all $X$ f second-order sort there exists a countable $\beta$-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-CA+BI+(Every true Π-formula is satisfied by a β-model of Δ-CA).M. Rathjen
Admissible proof theory and beyond
, Logic, Methodology and Philosophy of Science IX (Elsevier, 1994). Accessed 2022-12-04.

Additionally, there's a connection between β-models and the hyperjump, provably in ACA₀: 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

References

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Mathematical logic