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A MACHINE-ASSISTED PROOF OF GÖDEL’S INCOMPLETENESS THEOREMS FOR THE THEORY OF HEREDITARILY FINITE SETS

Published online by Cambridge University Press:  03 April 2014

LAWRENCE C. PAULSON
LAWRENCE C. PAULSON*
Affiliation:
University of Cambridge
*
*COMPUTER LABORATORY UNIVERSITY OF CAMBRIDGE CAMBRIDGE, CB3 0FD, UK E-mail: [email protected]
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Abstract

A formalization of Gödel’s incompleteness theorems using the Isabelle proof assistant is described. This is apparently the first mechanical verification of the second incompleteness theorem. The work closely follows Świerczkowski (2003), who gave a detailed proof using hereditarily finite set theory. The adoption of this theory is generally beneficial, but it poses certain technical issues that do not arise for Peano arithmetic. The formalization itself should be useful to logicians, particularly concerning the second incompleteness theorem, where existing proofs are lacking in detail.

Type
Research Article
Information
The Review of Symbolic Logic , Volume 7 , Issue 3 , September 2014 , pp. 484 - 498
Copyright
Copyright © Association for Symbolic Logic 2014 

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References

BIBLIOGRAPHY

Boolos, G. S. (1993). The Logic of Provability. Cambridge: Cambridge University Press.Google Scholar
de Bruijn, N. G. (1972). Lambda calculus notation with nameless dummies, a tool for automatic formula manipulation, with application to the Church-Rosser Theorem. Indagationes Mathematicae, 34, 381–392.Google Scholar
Feferman, S., editor (1986). Kurt Gödel: Collected Works, Vol. I, Oxford, UK: Oxford University Press.Google Scholar
Franzén, T. (2005). Gödel’s Theorem: An Incomplete Guide to Its Use and Abuse. Wellesley, Mass: A K Peters.CrossRefGoogle Scholar
Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I. Monatshefte für Mathematik und Physik, 38(1), 173–198.CrossRefGoogle Scholar
Kirby, L. (2007). Addition and multiplication of sets. Mathematical Logic Quarterly,53(1), 52–65.CrossRefGoogle Scholar
Nipkow, T., Paulson, L. C., & Wenzel, M. (2002). Isabelle/HOL: A Proof Assistant for Higher-Order Logic. Lecture Notes in Computer Science vol. 2283, Berlin: Springer.CrossRefGoogle Scholar
O’Connor, R. (2005). Essential incompleteness of arithmetic verified by Coq. In Hurd, J. &Melham, T., editors. TPHOLs, Lecture Notes in Computer Science vol. 3603, Berlin: Springer, pp. 245–260.Google Scholar
O’Connor, R. S. S. (2009). Incompleteness & Completeness: Formalizing Logic and Analysis in Type Theory. Ph. D. Thesis, Radboud University Nijmegen.Google Scholar
Paulson, L. C. (2013). A mechanised proof of Gödel’s incompleteness theorems using Nominal Isabelle. Submitted for publication.Google Scholar
Shankar, N. (1986). Proof-checking Metamathematics. Ph. D. Thesis, University of Texas , Austin.Google Scholar
Shankar, N. (1994). Metamathematics, Machines, and Gödel’s Proof. Cambridge : Cambridge University Press.CrossRefGoogle Scholar
Shankar, N. (2013). Shankar, Boyer, Church-Rosser and de Bruijn indices. E-mail.Google Scholar
Świerczkowski, S. (2003). Finite sets and Gödel’s incompleteness theorems. Dissertationes Mathematicae, 422, 1–58. Available from: http://journals.impan.gov.pl/dm/Inf/422-0-1.html.CrossRefGoogle Scholar
Urban, C., & Kaliszyk, C. (2012). General bindings and alpha-equivalence in Nominal Isabelle. Logical Methods in Computer Science, 8(2:14), 1–35.Google Scholar
Wenzel, M. (2007). Isabelle/Isar — a generic framework for human-readable proof documents. In Matuszewski, R. & Zalewska, A., editors. From Insight to Proof — Festschrift in Honour of Andrzej Trybulec, Studies in Logic, Grammar, and Rhetoric, vol. 10(23), University of Białystok.Google Scholar