Beta-model
In model theory, a mathematical discipline, a β-model (from the French "bon ordre", well-ordering[1]) is a model which is correct about statements of the form "X is well-ordered". The term was introduced by Mostowski (1959)[2][3] 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 set theory
It is 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 with parameters from M, iff .[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 am 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 .[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 , is equivalent to the statement "for all [of second-order sort], there exists a countable β-model M such that .[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 )[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, there's a connection between β-models and the hyperjump, provably in : for all sets of integers, has a hyperjump iff there exists a countable β-model such that .[4]p.251
References
- ^ C. Smoryński, "Nonstandard Models and Related Developments" (p. 189). From Harvey Friedman's Research on the Foundations of Mathematics (1985), Studies in Logic and the Foundations of Mathematics vol. 117.
- ^ a b K. R. Apt, W. Marek, "Second-order Arithmetic and Some Related Topics" (1973), p. 181
- ^ J.-Y. Girard, Proof Theory and Logical Complexity (1987), Part III: Π21-proof theory, p. 206
- ^ a b c S. G. Simpson, Subsystems of Second-Order Arithmetic (2009)
- ^ C. Mummert, S. G. Simpson, "An Incompleteness Theorem for βn-Models", 2004. Accessed 22 October 2023.
- ^ M. Rathjen, Proof theoretic analysis of KPM (1991), p.381. Archive for Mathematical Logic, Springer-Verlag. Accessed 28 February 2023.
- ^ M. Rathjen, Admissible proof theory and beyond , Logic, Methodology and Philosophy of Science IX (Elsevier, 1994). Accessed 2022-12-04.