Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-16T22:43:59.058Z Has data issue: false hasContentIssue false
  • English
  • Français

Constructive assertions in an extension of classical mathematics

Published online by Cambridge University Press:  12 March 2014

Vladimir Lifschitz
Vladimir Lifschitz*
Affiliation:
University of Texas at el Paso, El Paso, Texas 79968
Get access Rights & Permissions [Opens in a new window]

Extract

We distinguish between two kinds of mathematical assertions: objective and constructive. An objective assertion describes the universe of mathematical objects; a constructive one describes the (idealized) mathematician's ability to find mathematical objects with various properties. The familiar formalizations of classical mathematics are based on formal languages designed for expressing objective assertions only. The constructivist program stresses, on the contrary, the importance of constructive assertions; moreover, intuitionism claims that constructive activities of the mind constitute the very subject matter of mathematics, and thus questions the semantic status of objective assertions.

The purpose of this paper is to show that classical mathematics can be extended to include constructive sentences, so that both objective and constructive properties can be discussed in the framework of the same theory. To achieve this goal, we introduce a new property of mathematical objects, calculability.

The word “calculable” may be applied to objects of various types: natural numbers, integers, rational or real numbers, polynomials with rational or real coefficients, etc. In each case it has a different meaning, so that actually we define not one, but many new properties.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 47 , Issue 2 , June 1982 , pp. 359 - 387
Copyright
Copyright © Association for Symbolic Logic 1982

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]Bishop, E., Foundations of constructive analysts, McGraw-Hill, New York, 1967.Google Scholar
[2]Church, A., Introduction to mathematical logic. I, Princeton University Press, Princeton, N.J., 1956.Google Scholar
[3]Friedman, H., Axiomatic recursive function theory, Logic Colloquium '69, North-Holland, Amsterdam, 1971, pp. 113–137.CrossRefGoogle Scholar
[4]Barwise, Jon (Editor), Handbook of mathematical logic, North-Holland, Amsterdam, 1977.Google Scholar
[5]Heyting, A., Intuitionism, an introduction, North-Holland, Amsterdam, 1956.Google Scholar
[6]Kleene, S. C., Introduction to metamathematics, North-Holland, Amsterdam, Noordhoff, Groningen and Van Nostrand, New York, 1952.Google Scholar
[7]Kleene, S. C., Realizability: a retrospective survey, Cambridge Summer School in Mathematical Logic, (Mathias, A. R. D., Editor), Lecture Notes in Mathematics, vol. 337, Springer-Verlag, Berlin and New York, 1973, pp. 95–112.CrossRefGoogle Scholar
[8]Lifschitz, V., Calculable natural numbers (to appear).Google Scholar
[9]Markov, A. A., On constructive mathematics, Trudy Matematičeskogo Institute im V.A. Steklov, 67 (1962), pp. 8–14 (Russian). [English translation in American Mathematical Society Translations, vol. 98 (2) (1971), pp. 1–9]Google Scholar
[10]Troelstra, A. S. (Editor), Metamathematical investigation of intuitionistic arithmetic and analysis, Lecture Notes in Mathematics, vol. 344, Springer-Verlag, Berlin and New York, 1973.Google Scholar
[11]Rice, H. G., Recursive real numbers, Proceedings of the American Mathematical Society, vol. 5 (1954), pp. 784–791.CrossRefGoogle Scholar
[12]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[13]Shanin, N. A., On the constructive interpretation of mathematical judgments, Trudy Matematiceskogo Institute im V.A. Steklov, vol. 52 (1958), pp. 226–311 (Russian). [English translation in American Mathematical Society Translations, vol. 23 (2) (1963), pp. 109–189]Google Scholar
[14]van der Waerden, B. L., Algebra, Ungar, New York, 1970.Google Scholar