Abstract
Many efforts have been made in recent years to construct formal systems for mechanizing mathematical reasoning. A framework which seems particularly suitable for this task is ancestral logic – the logic obtained by augmenting first-order logic with a transitive closure operator. While the study of this logic has so far been mostly model-theoretical, this work is devoted to its proof theory (which is much more relevant for the task of mechanizing mathematics). We develop a Gentzen-style proof system TC G which is sound for ancestral logic, and prove its equivalence to previous systems for the reflexive transitive closure operator by providing translation algorithms between them. We further provide evidence that TC G indeed encompasses all forms of reasoning for this logic that are used in practice. The central rule of TC G is an induction rule which generalizes that of Peano Arithmetic (PA). In the case of arithmetics we show that the ordinal number of TC G is ε 0.
Access provided by Autonomous University of Puebla. Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Aho, A.V., Ullman, J.D.: Universality of data retrieval languages. In: Proceedings of the 6th ACM SIGACT-SIGPLAN Symposium on Principles of Programming Languages, pp. 110–119. ACM (1979)
Avron, A.: Transitive closure and the mechanization of mathematics. In: Kamareddine, F.D. (ed.) Thirty Five Years of Automating Mathematics. Applied Logic Series, vol. 28, pp. 149–171. Springer Netherlands (2003)
Avron, A.: Formalizing set theory as it is actually used. In: Asperti, A., Bancerek, G., Trybulec, A. (eds.) MKM 2004. LNCS, vol. 3119, pp. 32–43. Springer, Heidelberg (2004)
Avron, A.: A framework for formalizing set theories based on the use of static set terms. In: Avron, A., Dershowitz, N., Rabinovich, A. (eds.) Pillars of Computer Science. LNCS, vol. 4800, pp. 87–106. Springer, Heidelberg (2008)
Campbell, J.J.J.A., Reis, J.C.G.D., Wenzel, P.S.M., Sorge, V.: Intelligent computer mathematics (2008)
Constable, R.L., Allen, S.F., Bromley, H.M., Cleaveland, W.R., Cremer, J.F., Harper, R.W., Howe, D.J., Knoblock, T.B., Mendler, N.P., Panangaden, P., Sasaki, J.T., Smith, S.F.: Implementing Mathematics with the Nuprl Proof Development System. Prentice-Hall, Inc., Upper Saddle River (1986)
Ebbinghaus, H.-D., Flum, J.: Finite Model Theory, vol. 2. Springer (1995)
Fagin, R.: Generalized first-order spectra and polynomial-time recognizable sets (1974)
Gentzen, G.: Neue Fassung des Widerspruchsfreiheitsbeweises für die reine Zahlentheorie. Forschungen zur Logik 4, 19–44 (1969); English translation in: Szabo, M.E.: The collected work of Gerhard Gentzen. North-Holland, Amsterdam
Kamareddine, F.D.: Thirty five years of automating mathematics, vol. 28. Springer (2003)
Martin, R.M.: A homogeneous system for formal logic. The Journal of Symbolic Logic 8(1), 1–23 (1943)
Martin, R.M.: A note on nominalism and recursive functions. The Journal of Symbolic Logic 14(1), 27–31 (1949)
Momigliano, A., Tiu, A.: Induction and co-induction in sequent calculus. In: Berardi, S., Coppo, M., Damiani, F. (eds.) TYPES 2003. LNCS, vol. 3085, pp. 293–308. Springer, Heidelberg (2004)
Myhill, J.: A derivation of number theory from ancestral theory. The Journal of Symbolic Logic 17(3), 192–197 (1952)
Rudnicki, P.: An overview of the mizar project. In: Proceedings of the 1992 Workshop on Types for Proofs and Programs, pp. 311–330 (1992)
Shapiro, S.: Foundations without Foundationalism: A Case for Second-Order Logic: A Case for Second-Order Logic. Oxford University Press (1991)
Smith, P.: Ancestral arithmetic and isaacson’s thesis. Analysis 68(297), 1–10 (2008)
Tiu, A., Momigliano, A.: Cut elimination for a logic with induction and co-induction. Journal of Applied Logic 10(4), 330–367 (2012); Selected papers from the 6th International Conference on Soft Computing Models in Industrial and Environmental Applications
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cohen, L., Avron, A. (2014). Ancestral Logic: A Proof Theoretical Study. In: Kohlenbach, U., Barceló, P., de Queiroz, R. (eds) Logic, Language, Information, and Computation. WoLLIC 2014. Lecture Notes in Computer Science, vol 8652. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-44145-9_10
Download citation
DOI: https://doi.org/10.1007/978-3-662-44145-9_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-44144-2
Online ISBN: 978-3-662-44145-9
eBook Packages: Computer ScienceComputer Science (R0)