Beta-model

In model theory, a mathematical discipline, a β-model (from the French "bon ordre", well-ordering) is a model that 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 -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 with parameters from M, iff .^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 .

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 , is equivalent to the statement "for all [of second-order sort], there exists a countable β-model M such that .^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 ) is logically equivalent to the theory Δ-CA+BI+(Every true Π-formula is satisfied by a β-model of Δ-CA).

Additionally, proves a connection between β-models and the hyperjump: for all sets of integers, has a hyperjump iff there exists a countable β-model such that .^p. 251

Every β-model of comprehension is elementarily equivalent to an ω-model which is not a β-model.

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 such that the membership relations of is well-founded, and for any relation , " is well-founded" iff 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}

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