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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) as a strengthening of the notion of ω-model. In contrast to the notation for set-theoretic properties named by ordinals, such as ${\displaystyle \xi }$-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 Σ₁ formula ${\displaystyle \phi }$ with parameters from M, ${\displaystyle (\omega ,M,+,\times ,0,1,<)\vDash \phi }$ iff ${\displaystyle (\omega ,{\mathcal {P}}(\omega ),+,\times ,0,1,<)\vDash \phi }$. 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 ${\displaystyle n\geq 1}$.

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 ${\displaystyle X}$ , there exists a countable β-model M such that ${\displaystyle X\in M}$. (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 ${\displaystyle KPM}$) is logically equivalent to the theory Δ
₂-CA+BI+(Every true Π
₃-formula is satisfied by a β-model of Δ
₂-CA).

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 ${\displaystyle X\in M}$.

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 ${\displaystyle (M,{\mathcal {X}})}$ 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.

References

1. 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.
2. ^ K. R. Apt, W. Marek, "Second-order Arithmetic and Some Related Topics" (1973), p. 181
3. J.-Y. Girard, Proof Theory and Logical Complexity (1987), Part III: Π₂-proof theory, p. 206
4. ^ Simpson, Stephen G. (2009). Subsystems of second order arithmetic. Perspectives in logic. Association for Symbolic Logic (2nd ed.). Cambridge ; New York: Cambridge University Press. ISBN 978-0-521-88439-6. OCLC 288374692.
5. C. Mummert, S. G. Simpson, "An Incompleteness Theorem for β_n-Models", 2004. Accessed 22 October 2023.
6. M. Rathjen, Proof theoretic analysis of KPM (1991), p.381. Archive for Mathematical Logic, Springer-Verlag. Accessed 28 February 2023.
7. M. Rathjen, Admissible proof theory and beyond, Logic, Methodology and Philosophy of Science IX (Elsevier, 1994). Accessed 2022-12-04.
8. A. Mostowski, Y. Suzuki, "On ω-models which are not β-models". Fundamenta Mathematicae vol. 65, iss. 1 (1969).
9. K. J. Williams, "The Structure of Models of Second-order Set Theories", PhD thesis, 2018.

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