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Elementary intuitionistic theories

Published online by Cambridge University Press:  12 March 2014

C. Smorynski
C. Smorynski*
Affiliation:
University of Illinoisat Chicago Circle, Chicago, Illinois 60680
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Extract

The present paper concerns itself primarily with the decision problem for formal elementary intuitionistic theories and the method is primarily model-theoretic. The chief tool is the Kripke model for which the reader may find sufficient background in Fitting's book Intuitionistic logic model theory and forcing (North-Holland, Amsterdam, 1969). Our notation is basically that of Fitting, the differences being to favor more standard notations in various places.

The author owes a great debt to many people and would particularly like to thank S. Feferman, D. Gabbay, W. Howard, G. Kreisel, G. Mints, and R. Statman for their valuable assistance.

The method of elimination of quantifiers, which has long since proven its use in classical logic, has also been applied to intuitionistic theories (i) to demonstrate decidability ([9], [15], [17]), (ii) to prove the coincidence of an intuitionistic theory with its classical extension ([9], [17]), and (iii), as we will see below, to establish relations between an intuitionistic theory and its classical extension. The most general of these results is to be obtained from the method of Lifshits' quantifier elimination for the intuitionistic theory of decidable equality.

Since the details of Lifshits' proof have not been published, and since the proof yields a more general result than that stated in his abstract [15], we include the proof and several corollaries.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 38 , Issue 1 , March 1973 , pp. 102 - 134
Copyright
Copyright © Association for Symbolic Logic 1973

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References

REFERENCES

[1]Ackermann, W., Solvable cases of the decision problem, North-Holland, Amsterdam, 1954.Google Scholar
[2]Ershov, Yu. L., Lavrov, I. A., Taimanov, A. D. and Taitslin, M. A., Elementary theories, Russian Mathematical Surveys, vol. 20 (1965), pp. 35–106.CrossRefGoogle Scholar
[3]Gabbay, D., Sufficient conditions for the undecidability of intuitionistic theories, this Journal, vol. 37 (1972), pp. 375–384.Google Scholar
[4]Gabbay, D., Decidability of some intuitionistic predicate theories, this Journal, vol. 37 (1972), pp. 579–587.Google Scholar
[5]Heyting, A., Intuitionism; an introduction, 2nd edition, North-Holland, Amsterdam, 1966.Google Scholar
[6]de Jongh, D. H. J., The maximality of the intuitionistic predicate calculus with respect to Heyting's arithmetic, Compositio Mathematica (to appear).Google Scholar
[7]Kleene, S. C., Logical calculus and realizability, Acta Philosophica Fennica, vol. 18 (1965), pp. 71–80.Google Scholar
[8]Kreisel, G., Elementary completeness properties of intuitionistic logic with a note on negations of prenex formulae, this Journal, vol. 23 (1958), pp. 317–330.Google Scholar
[9]Kreisel, G., Notes concerning the elements of proof theory, Course Notes, University of California, Los Angeles.Google Scholar
[10]Kreisel, G. and Krivine, J. L., Elements of mathematical logic (model theory), North-Holland, Amsterdam, 1967.Google Scholar
[11]Kripke, S., Semantical analysis of intuitionistic logic. I, Formal systems and recursive functions (Crossley, J. N. and Dummett, M. A. E., Editors), North-Holland, Amsterdam, 1965.Google Scholar
[12]Kripke, S., Semantical analysis of intuitionistic logic. II (unpublished).Google Scholar
[13]Leivant, D., A note on translations of C into I, Mathematisch Centrum, Amsterdam, Technical Report, 1971.Google Scholar
[14]Lifshits, V. A., Deductive validity and reduction classes, Studies in constructive mathemattes and mathematical logic, Part 1 (Slisenko, A. O., Editor), Consultants Bureau, New York, 1969.Google Scholar
[15]Lifshits, V. A., Problem of decidability for some constructive theories of equalities, Studies in constructive mathematics and mathematical logic, Part 1 (Slisenko, A. O., Editor), Consultants Bureau, New York, 1969.Google Scholar
[16]Lifshits, V. A., Constructive mathematical theories consistent with classical logic, Problems in the constructive trend in mathematics. IV (Orevkov, V. P. and Shanin, N. A., Editors), American Mathematical Society, Providence, R.I., 1970.Google Scholar
[17]Lopez-Escobar, E. G. K., A decision method for the intuitionistic theory of successor, Indagationes Mathematicae, vol. 30 (1968), pp. 466–467.CrossRefGoogle Scholar
[18]Maslov, S. Yu., Mints, G. E. and Orevkov, V. P., Unsolvability in the constructive predicate calculus of certain classes of formulas containing only monadic predicate variables, Soviet Mathematics, Doklady, vol. 6 (1965), pp. 918–920.Google Scholar
[19]Mints, G. E. and Orevkov, V. P., An extension of the theorems of Glivenko and Kreisel to a certain class of formulas of predicate calculus, Soviet Mathematics, Doklady, vol. 4 (1963), pp. 1365–1367.Google Scholar
[20]Orevkov, V. P., On Glivenko sequent classes, The calculi of symbolic logic, American Mathematical Society, Providence, R.I., 1971.Google Scholar
[21]Rabin, M. O., Decidability of second order theories and automata on infinite trees, Transactions of the American Mathematical Society, vol. 141 (1969), pp. 1–35.Google Scholar
[22]Rogers, H. Jr., Certain logical reduction and decision problems, Annals of Mathematics, vol. 64 (1956), pp. 264–284.CrossRefGoogle Scholar
[23]Rose, G., Propositional calculus and realizability, Transactions of the American Mathematical Society, vol. 75 (1953), pp. 1–19.CrossRefGoogle Scholar
[24]Scott, D., Extending the topological interpretation to intuitionistic analysis, Compositio Mathematica, vol. 20 (1968), pp. 194–210.Google Scholar
[25]Scott, D., Extending the topological interpretation to intuitionistic analysis. II, Intuitionism and proof theory (Kino, A., Vesley, R. and Myhill, J., Editors), North-Holland, Amsterdam, 1970.Google Scholar
[26]Smorynski, C., The undecidability of some intuitionistic theories of equality and order this Journal, vol. 36 (1971), pp. 587–588.Google Scholar
[27]Tarski, A., Mostowski, A. and Robinson, R. M., Undecidable theories, North-Holland, Amsterdam, 1953.Google Scholar