Abstract
We describe a nonconstructive extension to primitive recursive arithmetic, both abstractly and as implemented on the Boyer-Moore prover. Abstractly, this extension is obtained by adding the unbounded µ operator applied to primitive recursive functions; doing so, one can define the Ackermann function and prove the consistency of primitive recursive arithmetic. The implementation does not mention the µ operator explicitly but has the strength to define the µ operator through the built-in functions EVAL$ and V&C$.
This is a preview of subscription content, log in via an institution to check access.
References
Aho, A. V., Hopcroft, J. E. and Ullman, J. D.: Data Structures and Algorithms, Addison-Wesley, 1983.
Beeson, M. L.: Foundations of Constructive Mathematics, Springer-Verlag, 1985.
Boyer, B. S., Goldschlag, D., Kaufmann, M. and Moore, J S.: Functional Instantiation in First Order Logic, in Artificial Intelligence and Mathematical Theory of Computation: Papers in Honor of John McCarthy, V. Lifschitz (ed.), Academic Press, 1991, pp. 7–26.
Boyer, R. S. and Moore, J S.: The addition of bounded quantification and partial functions to a computational logic and its theorem prover, J. Automated Reasoning 4 (1988), 117–172.
Boyer, R. S. and Moore, J S.: A Computational Logic Handbook, Academic Press, 1988.
Brock, B. C. and Hunt, W. A.: An Overview of the Formal Specification and Verification of the FM9001 Microprocessor, preprint, currently available on WWW at http://www.cli.com/hardware/fm9001.html.
Gentzen, G.: Die Widerspruchsfreiheit der reinen Zahlentheorie, Mathematische Annalen 112 (1936), 493–565.
Gödel, K.: Zur intuitionistischen Arithmetik und Zahlentheorie, Ergebnisse eines Mathematischen Kolloquiums 4 (1933), 34–38, reprinted in Feferman, Dawson, Kleene, Moore, Solovay, and van Heijenoort, Kurt Gödel Collected Works Vol. 1, Oxford University Press, 1986.
Goodstein, R. L.: Recursive Number Theory, North-Holland 1964.
Kaufmann, M.: An extension of the Boyer-Moore theorem prover to support first-order quantification, J. Automated Reasoning 9 (1992), 355–372.
Ketonen, J. and Solovay, S.: Rapidly growing Ramsey functions, Annals of Math. 113 (1981), 267–314.
Kleene, S. C.: Introduction to Metamathematics, Van Nostrand, 1952.
Kunen, K.: A Ramsey theorem in Boyer-Moore logic, J. Automated Reasoning 15 (1995), 217–235.
Paris, J. and Harrington, L.: A mathematical incompleteness in Peano arithmetic, in Handbook of Mathematical Logic, J. Barwise (ed.), North-Holland, 1978, pp. 1133–1142.
Parsons, C.: On a Number-theoretic choice scheme and its relation to induction, in Intuitionism and Proof Theory, Kino, Myhill, and Vessley (eds), North-Holland, pp. 459–473. See also JSL 37 (1972), 466–482.
Sieg, W.: Fragments of arithmetic, APAL 28 (1985), 33–71.
Troelstra, A. S.: Constructivism in Mathematics Vol. 1, North-Holland, 1988.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kunen, K. Nonconstructive Computational Mathematics. Journal of Automated Reasoning 21, 69–97 (1998). https://doi.org/10.1023/A:1005888712422
Issue Date:
DOI: https://doi.org/10.1023/A:1005888712422