Beta-model
In model theory, a mathematical discipline, a β-model is a model that is correct about statements of the form "X is well-ordered". The term was introduced by Mostowski 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 βn-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 , there exists a countable β-model M such that.p. 253 Also, the theory extending KP with a canonical axiom schema for a recursively Mahlo universe is logically equivalent to the theory Δ-CA+BI+.
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.