Abstract
This paper deals with a proof theory for a theory T N of \(\Pi _{N}\)-reflecting ordinals using a system \(\mathit{Od}(\Pi _{N})\) of ordinal diagrams. This is a sequel to the previous one (Arai, Ann Pure Appl Log 129:39–92, 2004) in which a theory for \(\Pi _{3}\)-reflecting ordinals is analysed proof-theoretically.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For simplicity we suppress the parameter. Correctly \(\forall u(A(u)\, \rightarrow \,\exists z(u < z\,\&\,A^{z}(u)))\).
References
T. Arai, A system (O(π n ), < ) of ordinal diagrams. Handwritten note (1994)
T. Arai, Proof theory for reflecting ordinals IV: \(\Pi _{N}\)-reflecting ordinals. Handwritten note (1994)
T. Arai, Proof theory for theories of ordinals I: recursively Mahlo ordinals. Ann. Pure Appl. Log. 122, 1–85 (2003)
T. Arai, Proof theory for theories of ordinals II: \(\Pi _{3}\)-reflection. Ann. Pure Appl. Log. 129, 39–92 (2004)
T. Arai, Iterating the recursively Mahlo operations, in Proceedings of the Thirteenth International Congress of Logic Methodology, Philosophy of Science, ed. by C. Glymour, W. Wei, D. Westerstahl (College Publications, London, 2009), pp. 21–35
T. Arai, Wellfoundedness proofs by means of non-monotonic inductive definitions II: first order operators. Ann. Pure Appl. Log. 162, 107–143 (2010)
G. Gentzen, Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie, Forschungen zur Logik und zur Grundlegung der exakter Wissenschaften. Neue Folge 4, 19–44 (1938)
M. Rathjen, Proof theory of reflection. Ann. Pure Appl. Log. 68, 181–224 (1994)
M. Rathjen, An ordinal analysis of stability. Arch. Math. Log. 44, 1–62 (2005)
M. Rathjen, An ordinal analysis of parameter free \(\pi _{2}^{1}\)-comprehension. Arch. Math. Log. 44, 263–362 (2005)
W.H. Richter, P. Aczel, Inductive definitions and reflecting properties of admissible ordinals, in Generalized Recursion Theory, ed. by J.E. Fenstad, P.G. Hinman (North-Holland, Amsterdam, 1974), pp. 301–381
G. Takeuti, Proof Theory, 2nd edn. (North-Holland, Amsterdam, 1987). Reprinted from Dover Publications, 2013
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Arai, T. (2015). Proof Theory for Theories of Ordinals III: \(\Pi _{N}\) -Reflection. In: Kahle, R., Rathjen, M. (eds) Gentzen's Centenary. Springer, Cham. https://doi.org/10.1007/978-3-319-10103-3_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-10103-3_14
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-10102-6
Online ISBN: 978-3-319-10103-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)