Abstract
We introduce and analyze two theories for typed (accessible part) inductive definitions and establish their proof-theoretic ordinal to be the small Veblen ordinal \(\vartheta \Omega ^\omega \). We investigate on the one hand the applicative theory \(\mathsf {FIT}\) of functions, (accessible part) inductive definitions, and types. It includes a simple type structure and is a natural generalization of S. Feferman’s system \(\mathrm {QL}(\mathsf {F}_\mathsf {0}\text {-}\mathsf {IR}_{N})\). On the other hand, we investigate the arithmetical theory \(\mathsf {TID}\) of typed (accessible part) inductive definitions, a natural subsystem of \(\mathsf {ID}_1\), and carry out a wellordering proof within \(\mathsf {TID}\) that makes use of fundamental sequences for ordinal notations in an ordinal notation system based on the finitary Veblen functions. The essential properties for describing the ordinal notation system are worked out.
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References
Afshari, B., Rathjen, M.: A note on the theory of positive induction, \({{\rm ID}}^\ast _1\). Arch. Math. Logic 49, 275–281 (2010). https://doi.org/10.1007/s00153-009-0168-9
Beeson, M.J.: Foundations of Constructive Mathematics: Metamathematical studies. Springer, Berlin (1985)
Buchholz, W., Feferman, S., Pohlers, W., Sieg, W.: Iterated Inductive Definitions and Subsystems of Analysis: Recent Proof-Theoretical Studies, volume 897 of Lecture Notes in Mathematics. Springer, Berlin (1981)
Bridge, J.: A simplification of the Bachmann method for generating large countable ordinals. J. Symb. Logic 40(2), 171–185 (1975)
Buchholz, W., Schütte, K.: Proof Theory of Impredicative Subsystems of Analysis, volume 2 of Studies in Proof Theory Monographs. Bibliopolis, Napoli (1988)
Buchholz, W.: Prädikative Beweistheorie (Predicative proof theory). Lecture notes, University of Munich (2004–2005)
Buchholz, W.: A survey on ordinal notations around the Bachmann–Howard ordinal. In: Kahle, R., Strahm, T., Studer, T. (eds.) Advances in Proof Theory, pp. 1–29. Birkhaeuser, Springer, Basel (2016)
Cantini, A.: On the relation between choice and comprehension principles in second order arithmetic. J. Symb. Logic 51(2), 360–373 (1986)
Feferman, S.: Constructive theories of functions and classes. In: van Dalen, D., Boffa, M., Mcaloon, K. (eds.) Logic Colloquium ’78 Proceedings of the colloquium held in Mons, volume 97 of Studies in Logic and the Foundations of Mathematics, pp. 159–224. Elsevier (1979)
Feferman, S.: Logics for termination and correctness of functional programs, II. Logics of strength PRA. In: Aczel, P., Simmons, H., Wainer, S.S. (eds.) Proof Theory, pp. 195–225. Cambridge University Press, Cambridge (1992)
Feferman, S., Jäger, G., Strahm, T.: Foundations of Explicit Mathematics (in preparation)
Jäger, G.: Metapredicative and explicit Mahlo: a proof-theoretic perspective. In: Cori, R., Razborov, A., Todorcevic, S., Wood, C. (eds.) Proceedings of Logic Colloquium ’00, volume 19 of Association of Symbolic Logic Lecture Notes in Logic, pp. 272–293. AK Peters (2005)
Jäger, G., Kahle, R., Setzer, A., Strahm, T.: The proof-theoretic analysis of transfinitely iterated fixed point theories. J. Symb. Logic 64(1), 53–67 (1999)
Jäger, G., Strahm, T.: Bar induction and \(\omega \) model reflection. Ann. Pure Appl. Logic 97(1–3), 221–230 (1999)
Jäger, G., Strahm, T.: Reflections on reflections in explicit mathematics. Ann. Pure Appl. Logic 136(1–2), 116–133 (2005) (Festschrift on the occasion of Wolfram Pohlers’ 60th birthday)
Probst, D.: The proof-theoretic analysis of transfinitely iterated quasi least fixed points. J. Symb. Logic 71(3), 721–746 (2006)
Probst, D.: Modular Ordinal Analysis of Subsystems of Second-Order Arithmetic of Strength up to the Bachmann–Howard Ordinal. Habilitation, Universität Bern (2017)
Ranzi, F.: From a flexible type system to metapredicative wellordering proofs. PhD thesis, Universität Bern. http://www.iam.unibe.ch/ltgpub/2015/ran15.pdf (2015). https://doi.org/10.7892/boris.75102
Rathjen, M.: Fragments of Kripke–Platek set theory with infinity. In: Aczel, P., Simmons, H., Wainer, S.S. (eds.) Proof Theory. A Selection of Papers from the Leeds Proof Theory Programme 1990, pp. 251–273. Cambridge University Press, Cambridge (1992)
Rathjen, M., Weiermann, A.: Proof-theoretic investigations on Kruskal’s theorem. Ann. Pure Appl. Logic 60, 49–88 (1993)
Schütte, K.: Kennzeichnung von Ordnungszahlen durch rekursiv erklärte Funktionen. Math. Ann. 127, 15–32 (1954). https://doi.org/10.1007/BF01361109
Schütte, K.: Beziehungen des Ordinalzahlensystems \({{\rm OT}}(\vartheta )\) zur Veblen-Hierarchie. Unpublished notes (1992)
Simpson, S.G.: Subsystems of Second Order Arithmetic, 2nd edn. Cambridge University Press, Cambridge (2009) (Cambridge Books Online)
Strahm, T.: First steps into metapredicativity in explicit mathematics. In: Barry Cooper, S., Truss, J.K. (eds.) Sets and Proofs, volume 258 of London Mathematical Society Lecture Notes, pp. 383–402. Cambridge University Press, Cambridge (1999)
Van der Meeren, J., Rathjen, M., Weiermann, A.: An order-theoretic characterization of the Howard–Bachmann-hierarchy. Arch. Math. Logic 56(1–2), 79–118 (2017)
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Florian Ranzi’s research was supported by the Swiss National Science Foundation.
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Ranzi, F., Strahm, T. A flexible type system for the small Veblen ordinal. Arch. Math. Logic 58, 711–751 (2019). https://doi.org/10.1007/s00153-019-00658-x
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DOI: https://doi.org/10.1007/s00153-019-00658-x
Keywords
- Proof theory
- Inductive definitions
- Applicative theories
- Small Veblen ordinal
- Finitary Veblen functions
- Metapredicativity
- Higher types
- Subsystems of second-order arithmetic