Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-23T13:11:48.965Z Has data issue: false hasContentIssue false
  • English
  • Français

Predicate Logics of Constructive Arithmetical Theories

Published online by Cambridge University Press:  12 March 2014

Albert Visser
Albert Visser*
Affiliation:
Department of Philosophy, Utrecht University, Heidelberglaan 8, 3584 CS Utrecht, The Netherlands, E-mail: Albert.Visser@phil.uu.nl
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we show that the predicate logics of consistent extensions of Heyting's Arithmetic plus Church's Thesis with uniqueness condition are complete . Similarly, we show that the predicate logic of HA*. i.e. Heyting's Arithmetic plus the Completeness Principle (for HA*) is complete . These results extend the known results due to Valery Plisko. To prove the results we adapt Plisko's method to use Tennenbaum's Theorem to prove ‘categoricity of interpretations’ under certain assumptions.

Keywords

Relative interpretationspredicate logics of arithmetical theoriesconstructive logic
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 71 , Issue 4 , December 2006 , pp. 1311 - 1326
Copyright
Copyright © Association for Symbolic Logic 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Artëmov, S.N., Arithmetically complete modal theories, Semiotika i informatika, 14, VINITI, Moscow, 1980, In Russian. English translation in: American Mathematical Society Translations (2), 135: 39–54, 1987, pp. 115–133.Google Scholar
[2]Artemov, S.N. and Beklemishev, L.D., Provability logic, Handbook of Philosophical Logic, 2nded. (Gabbay, D. and Guenthner, F., editors), vol. 13, Kluwer, Dordrecht, 2004, pp. 229–403.Google Scholar
[3]Beeson, M., The nonderivability in intuitionistic formal systems of theorems on the continuity of effective operations, this Journal, vol. 40 (1975). pp. 321–346.Google Scholar
[4]Boolos, G., The logic of provability, Cambridge University Press, Cambridge, 1993.Google Scholar
[5]Boolos, G. and McGee, V., The degree of the set of sentences of predicate provability logic that are true under every interpretation, this Journal, vol. 52 (1987), no. 1, pp. 165–171.Google Scholar
[6]Burr, W., Fragments of Heyting Arithmetic, this Journal, vol. 65(3) (2000), pp. 1223–1240.Google Scholar
[7]de Jongh, D.H.J. and Visser, A., Embeddings of Heyting algebras, Logic: from foundations to applications (Hodges, W., Hyland, M., Steinhorn, C., and Truss, J., editors), Clarendon Press, Oxford, 1996, pp. 187–213.CrossRefGoogle Scholar
[8]Lifschitz, V., CT0 is stronger than CT0!, Proceedings of the American Mathematical Society, vol.73 (1979), pp. 101–106.Google Scholar
[9]McCarty, D.C., Constructive validity is nonarithmetic, this Journal, vol. 53 (1988), pp. 1036–1041.Google Scholar
[10]McCarty, D.C., Incompleteness in intuitionistic metamathematics, The Notre Dame Journal of Formal Logic, vol. 32 (1991), pp. 323–358.CrossRefGoogle Scholar
[11]Montagna, F., The predicate modal logic of provability, Notre Dame Journal of Formal Logic, vol. 25 (1987), pp. 179–189.Google Scholar
[12]van Oosten, J., Lifschitz' readability, this Journal, vol. 55 (1990), pp. 805–821.Google Scholar
[13]Plisko, V. E., On realizable predicative formulas, Soviet Math. Dokl., vol. 14 (1973), pp. 1420–1423.Google Scholar
[14]Plisko, V. E., The nonarithmeticity of the class of realizable formulas, Mathematics of the USSR: Izvestiya, vol. 11 (1977), pp. 453–471.CrossRefGoogle Scholar
[15]Plisko, V. E.. Some variants of the notion of realizability for predicate formulas, Mathematics of the USSR: Izvestiya, vol. 12 (1978), pp. 588–604.CrossRefGoogle Scholar
[16]Plisko, V. E., Absolute realizability of predicate formulas, Mathematics of the USSR: Izvestiya, vol. 22 (1983), pp. 291–308.CrossRefGoogle Scholar
[17]Plisko, V. E., Modified realizability and predicate logic, Abstracts of the All Union Conference in mathematical logic, Alma Ata, 1990, (1990), Russian.Google Scholar
[18]Plisko, V. E., Constructive formalization of the Tennenbaum theorem, Mathematical Notes, (1991), pp. 950–957. Translated from Matematicheskie Zametki, vol. 48, pp. 108–118, 1990.Google Scholar
[19]Plisko, V. E., On arithmetic complexity of certain constructive logics, Mathematical Notes, (1993), pp. 701–709, Translated from Matematicheskie Zametki, vol. 52, pp. 94–104, 1992.Google Scholar
[20]Plisko, V. E., Arithmetic complexity of the predicate logics of certain complete arithmetic theories, Annals of Pure and Applied Logic, vol. 113 (2002), pp. 243–259.CrossRefGoogle Scholar
[21]Troelstra, A.S. and van Dalen, D., Constructivism in Mathematics, vol 1, Studies in Logic and the Foundations of Mathematics, vol. 121, North Holland, Amsterdam, 1988.Google Scholar
[22]Vardanyan, V.A., Arithmetic complexity of predicate logics of provability and their fragments, Doklady Akademii Nauk SSSR, vol. 288 (1986), no. 1, pp. 11–14, In Russian. English translation in Soviet Mathematics Doklady 33:569–572, 1986.Google Scholar
[23]Visser, A., On the completeness principle, Annals of Mathematical Logic, vol. 22 (1982), pp. 263–295.CrossRefGoogle Scholar
[24]Visser, A., Rules and arithmetics, The Notre Dame Journal of Formal Logic, vol. 40 (1999), no. 1, pp. 116–140.CrossRefGoogle Scholar
[25]Visser, A., Substitutions of -sentences: explorations between intuitionistic propositional logic and intuitionistic arithmetic, Annals of Pure and Applied Logic, vol. 114 (2002), pp. 227–271.CrossRefGoogle Scholar
[26]Visser, A., Faith & falsity: a study of faithful interpretations and false -sentences, Annals of Pure and Applied Logic, vol. 131 (2005), pp. 103–131.CrossRefGoogle Scholar
[27]Visser, A., Categories of Theories and Interpretations, Logic in Tehran (Enayat, Ali, Kalantari, Iraj, and Moniri, Mojtaba, editors), Lecture Notes in Logic, vol. 26, ASL and A.K. Peters, Ltd., Wellesley, Mass., 2006, pp. 284–341.CrossRefGoogle Scholar
[28]Visser, A. and de Jonge, M., No escape from Vardanyan's Theorem, Archive for Mathematical Logic, vol. 45 (2006), no. 5, pp. 539–554.CrossRefGoogle Scholar
[29]Yavorsky, R.E., Logical schemes for first order theories, Logical foundations of computer science, LFCS '97 (Adian, Sergei I. and Nerode, Anil, editors), Lecture Notes in Computer Science, vol. 1234, Springer, 1997, pp. 410–418.CrossRefGoogle Scholar