Abstract
Standard formalizations of constructive mathematics (’constructive’ here in the narrow sense of Bishop (1967): choice sequences are regarded as inacceptable, and Church’s thesis is not assumed) can be carried out in formal systems based on intuitionistic logic which become classical formal systems on addition of the principle of the excluded third. The fact that in such systems for constructive mathematics the logical operations permit an interpretation different from the classical truth-functional one is then solely expressed by the fact that less axioms are assumed. As is well-known, this results in formal properties1 such as.
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Bibliography
Beeson, M. J.: 1972, Metamathematics of Constructive Theories of Effective Operations, Thesis, Stanford University. (See also Beeson, 1975.).
Beeson, M. J.: 1975, ‘The Nonderivability in Intuitionistic Formal Systems of Theorems on the Continuity of Effective Operations’, Journ. Symbolic Logic 40, 321–346.
Beeson, M. J.: A,’ The Unprovability in Constructive formal Systems of the Continuity of Effective Operations on the Reals’. To appear.
Beeson, M. J.: B, ‘Principles of Continuous Choice and Continuity of Functions in Formal Systems for Constructive Mathematics’. To appear.
Bishop, E.: 1967, Foundations of Constructive Analysis, McGraw Hill, New York.
Čeitin, G. S.: 1959, ‘Algorithmic Operators in Constructive Complete Separable Metric Spaces’ (Russian), Doklady Akad. Nauk 128, 49–52.
Dalen, D. van: 1973, ‘Lectures on Intuitionism’, in A. R. D. Mathias, H. Rogers (eds.), Cambridge Summer School in Mathematical Logic, Springer-Verlag, Berlin, pp. 1–94.
Dalen, D. van: 1974A, ‘Choice Sequences in Beth models’, Notes on Logic and Computer Science 20, Dept. of Mathematics, Rijksuniversiteit, Utrecht (The Netherlands). Revised in ‘An Interpretation of Intuitionistic Analysis’, to appear in Annals of Math. Logic.
Dalen, D. van: 1975, ‘Experiments in Lawlessness’, Notes on Logic and Computer Science 28, Dept. of Mathematics, Rijksuniversiteit, Utrecht (The Netherlands). Revised in ‘An Interpretation of Intuitionistic Analysis’, to appear in Annals of Math. Logic.
Dalen, D. van: 1975A, ‘The Use of Kripke’s Schema as a Reduction Principle’, Preprint 11, Dept. of Mathematics, University of Utrecht.
Dragalin, A. G.: 1973, ‘Constructive Mathematics and Models of Intuitionistic Theories’, in P. Suppes, L. Henkin, Gr. C. Moisil, A. Joja (eds.), Logic, Methodology and Philosophy of Science IV, North-Holland Publ. Co., Amsterdam, 111–128.
Friedman, H.: 1975, ‘Set Theoretic Foundations for Constructive Analysis and the Hilbert Program’, Manuscript, Dept. of Mathematics, State University of New York at Buffalo, Buffalo, N.Y.
Jongh, D. H. J. de and Smorynski, C. A.: 1974, ‘Kripke Models and the Theory of Species’ Report 74-03, Dept. of Mathematics, Univ. of Amsterdam. Appeared in: Annals of Mathematical Logic 9 (1976), 157–186.
Kleene, S. C: 1965, ‘Logical Calculus and Realizability’, Acta Philosophica Fennica 18, 71–80.
Kleene, S. C. and Vesley, R. E.: 1965, The Foundations of Intuitionistic Mathematics, Especially in Relation to Recursive Functions, North-Holland Publ. Co., Amsterdam.
Kreisel, G.: 1968, ‘Lawless Sequences of Natural Numbers’, Compositio Math. 20, 222–248.
Kreisel, G.: 1971, ‘A Survey of Proof Theory II’, in J.-E. Fenstad (ed.) Proceedings of the Second Scandinavian Logic Symposium, North-Holland Publ. Co., Amsterdam, pp. 109–170.
Kreisel, G.: 1972, ‘Which Number-Theoretic Problems can be Solved in Recursive Progressions on Paths Through 0?’, J. Symbolic Logic 37, 311–344.
Kreisel, G. and Troelstra, A. S.: 1970, ‘Formal Systems for Some Branches of Intuitionistic Analysis’, Annals of Math. Logic 1, 229–387.
Moschovakis, J. R.: 1973, ‘A Topological Interpretation of Second-order Intuitionistic Arithmetic’, Compositio Math. 26, 261–275.
Moschovakis, Y. N.: 1964, ‘Recursive Metric Spaces’, Fund. Math. 55, 215–238.
Orevkov, V. P.: 1963, ‘A Constructive Map of the Square into itself, which Moves every Constructive Point’ (Russian), Doklady Akad. Nauk SSSR 152, 55–58; translated in Soviet Mathematics 4 (1963), 1253-1256.
Orevkov, V. P.: 1964, ‘On Constructive Mappings of a Circle into itself (Russian)’, Trudy Mat. Inst. Steklov 72, pp. 437–461; translated in Translations AMS 100, 69-100.
Troelstra, A. S.: 1969, Principles of Intuitionism, Springer-Verlag, Berlin.
Troelstra, A. S. (ed.): 1973, Metamathematical Investigation of Intuitionistic Arithmetic and Analysis, Springer-Verlag, Berlin.
Troelstra, A. S.: 1973A, ‘Notes on Intuitionistic Second-Order Arithmetic’, in A. R. D. Mathias, H. Rogers (eds.), Cambridge Summer School in Mathematical Logic, Springer-Verlag, Berlin, 171–205.
Troelstra, A. S.: 1976, Choice Sequences, a Chapter of Intuitionistic Mathematics, Oxford University Press, Oxford. To appear.
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© 1977 D. Reidel Publishing Company, Dordrecht, Holland
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Troelstra, A.S. (1977). Axioms for Intuitionistic Mathematics Incompatible with Classical Logic. In: Butts, R.E., Hintikka, J. (eds) Logic, Foundations of Mathematics, and Computability Theory. The University of Western Ontario Series in Philosophy of Science, vol 9. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1138-9_4
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