Beta-model

{{Short description|A class of "well-behaved" models in set theory}}

In model theory, a mathematical discipline, a β-model (from the French "bon ordre", well-orderingC. Smoryński, "[https://www.sciencedirect.com/science/article/abs/pii/S0049237X09701600 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.) 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, "[https://www.sciencedirect.com/science/article/pii/0003484374900011 Second-order Arithmetic and Some Related Topics]" (1973), p. 181J.-Y. Girard, Proof Theory and Logical Complexity (1987), Part III: Π₂¹-proof theory, p. 206 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 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 $\phi$ with parameters from M, $(\omega,M,+,\times,0,1,<)\vDash\phi$ iff $(\omega,\mathcal P(\omega),+,\times,0,1,<)\vDash\phi$ .{{Cite book |last=Simpson |first=Stephen G. |url=https://www.worldcat.org/title/288374692 |title=Subsystems of second order arithmetic |date=2009 |publisher=Cambridge University Press |others=Association for Symbolic Logic |isbn=978-0-521-88439-6 |edition=2nd |series=Perspectives in logic |location=Cambridge; New York |oclc=288374692}}^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 $n\geq 1$ .C. Mummert, S. G. Simpson, "[https://sgslogic.net/t20/papers/betan.pdf An Incompleteness Theorem for β_n-Models]", 2004. Accessed 22 October 2023.

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 $\mathsf{ATR}_0$ , $\Pi^1_1\mathsf{-CA}_0$ is equivalent to the statement "for all $X$ [of second-order sort], there exists a countable β-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 $KPM$ )M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/KPM-Ordinal-Analysis.pdf Proof theoretic analysis of KPM] (1991), p.381. Archive for Mathematical Logic, Springer-Verlag. Accessed 28 February 2023. is logically equivalent to the theory Δ{{su|b=2|p=1}}-CA+BI+(Every true Π{{su|b=3|p=1}}-formula is satisfied by a β-model of Δ{{su|b=2|p=1}}-CA).M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/BEYOND.pdf Admissible proof theory and beyond], Logic, Methodology and Philosophy of Science IX (Elsevier, 1994). Accessed 2022-12-04.

Additionally, $\mathsf{ACA}_0$ proves a connection between β-models and the hyperjump: 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

Every β-model of comprehension is elementarily equivalent to an ω-model which is not a β-model.A. Mostowski, Y. Suzuki, "[https://eudml.org/doc/214123 On ω-models which are not β-models]". Fundamenta Mathematicae vol. 65, iss. 1 (1969).

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 $(M, \mathcal X)$ such that the membership relations of $(M, \mathcal X)$ is well-founded, and for any relation $R\in\mathcal X$ , $(M, \mathcal X)\vDash$ " $R$ is well-founded" iff $R$ is in fact well-founded. While there is no least transitive model of MK, there is a least β-model of MK.K. J. Williams, "[https://arxiv.org/pdf/1804.09526.pdf The Structure of Models of Second-order Set Theories]", PhD thesis, 2018.^{pp. 17,154–156}

References

Category:Mathematical logic