Abstract
We present a first-order theory of ordinals without resorting to set theory. The theory is implemented in the KeY program verification system which is in turn used to prove termination of a Java program computing the Goodstein sequences.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ahrendt, W., Beckert, B., Bubel, R., Hähnle, R., Schmitt, P.H., Ulbrich, M. (eds.): Deductive Software Verification - The KeY Book - From Theory to Practice. LNCS, vol. 10001. Springer, Cham (2016). doi:10.1007/978-3-319-49812-6
Bachmann, H.: Transfinite Zahlen. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 1, 2nd edn. Springer, Heidelberg (1967). doi:10.1007/978-3-642-88514-3
Belinfante, J.G.F.: On computer-assisted proofs in ordinal number theory. J. Autom. Reason. 22(2), 341–378 (1999)
Blanchette, J.C., Fleury, M., Traytel, D.: Nested multisets, hereditary multisets, and syntactic ordinals in isabelle/hol (under submission)
Blanchette, J.C., Fleury, M., Traytel, D.: Formalization of nested multisets, hereditary multisets, and syntactic ordinals. Archive of Formal Proofs, November 2016. http://isa-afp.org/entries/Nested_Multisets_Ordinals.shtml. Formal proof development
Castéran, P., Contejean, E.: On ordinal notations. https://github.com/coq-contribs/cantor
Castéran, P., Contejean, E.: On ordinal notations. https://coq.inria.fr/V8.2pl1/contribs/Cantor.epsilon0.Goodstein.html
Grimm, J.: Implementation of three types of ordinals in Coq. Research report RR-8407, CRISAM - Inria Sophia Antipolis (2013)
Huffman, B.: Countable ordinals. Archive of Formal Proofs, November 2005. http://afp.sf.net/entries/Ordinal.shtml. Formal proof development
Kirby, L., Paris, J.: Accessible independence results for Peano arithmetic. Bull. Lond. Math. Soc. 14(4), 285–293 (1982)
Klaua, D.: Kardinal- und Ordinalzahlen, Teil 1. Wissenschaftliche Taschenbücher: Mathematik, Physik. Vieweg Braunschweig (1974)
Klaua, D.: Kardinal- und Ordinalzahlen, Teil 2. Wissenschaftliche Taschenbücher: Mathematik, Physik. Vieweg Braunschweig (1974)
Manolios, P., Vroon, D.: Algorithms for ordinal arithmetic. In: Baader, F. (ed.) CADE 2003. LNCS (LNAI), vol. 2741, pp. 243–257. Springer, Heidelberg (2003). doi:10.1007/978-3-540-45085-6_19
Manolios, P., Vroon, D.: Ordinal arithmetic: algorithms and mechanization. J. Autom. Reason. 34(4), 387–423 (2005)
Norrish, M., Huffman, B.: Ordinals in HOL: transfinite arithmetic up to (and beyond) \(\omega _1\). In: Blazy, S., Paulin-Mohring, C., Pichardie, D. (eds.) ITP 2013. LNCS, vol. 7998, pp. 133–146. Springer, Heidelberg (2013). doi:10.1007/978-3-642-39634-2_12
Rathjen, M.: Goodstein revisited. ArXiv e-prints, May 2014
Rathjen, M.: Goodstein’s theorem revisited. In: Kahle, R., Rathjen, M. (eds.) Gentzen’s Centenary, pp. 229–242. Springer, Cham (2015). doi:10.1007/978-3-319-10103-3_9
Goodstein, R.L.: On the restricted ordinal theorem. JSL 9, 33–41 (1944)
Schmitt, P.H.: A first-order theory of ordinals. Technical report 6, Department of Informatics, Karlsruhe Institute of Technology (2017)
Takeuti, G.: A formalization of the theory of ordinal numbers. J. Symb. Logic 30, 295–317 (1965)
Takeuti, G., Zaring, W.M.: Introduction to Axiomatic Set Theory. Graduate Texts in Mathematics, vol. 1. Springer, New York (1971). doi:10.1007/978-1-4684-9915-5
Ulbrich, M.: Dynamic logic for an intermediate language: verification, interaction and refinement. Ph.D. thesis, Karlsruhe Institute of Technology, June 2013
Winkler, S., Zankl, H., Middeldorp, A.: Beyond Peano arithmetic–automatically proving termination of the goodstein sequence. In: van Raamsdonk, F. (ed.) 24th International Conference on Rewriting Techniques and Applications (RTA 2013). Leibniz International Proceedings in Informatics (LIPIcs), Dagstuhl, Germany, vol. 21, pp. 335–351. Schloss Dagstuhl–Leibniz-Zentrum fuer Informatik (2013)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this paper
Cite this paper
Schmitt, P.H. (2017). A Mechanizable First-Order Theory of Ordinals. In: Schmidt, R., Nalon, C. (eds) Automated Reasoning with Analytic Tableaux and Related Methods. TABLEAUX 2017. Lecture Notes in Computer Science(), vol 10501. Springer, Cham. https://doi.org/10.1007/978-3-319-66902-1_20
Download citation
DOI: https://doi.org/10.1007/978-3-319-66902-1_20
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-66901-4
Online ISBN: 978-3-319-66902-1
eBook Packages: Computer ScienceComputer Science (R0)