All the mathematics in the world: logical validity and classical set theory
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Published:
Dec 28, 2017
Keywords:
ustification of deduction set validity class validity general validity intuitionism topological models
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Abstract
A recognizable topological model construction shows that any consistent principles of classical set theory, including the validity of the law of the excluded third, together with a standard class theory, do not suffice to demonstrate the general validity of the law of the excluded third. This result calls into question the classical mathematician's ability to offer solid justifications for the logical principles he or she favors.
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McCarty, D. C. (2017). All the mathematics in the world: logical validity and classical set theory. Philosophical Problems in Science (Zagadnienia Filozoficzne W Nauce), (63), 5–29. Retrieved from https://zfn.edu.pl/index.php/zfn/article/view/394
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References
Aczel, P., 1978. The type-theoretic interpretation of constructive set theory. In: A. MacIntyre et al. (eds.), Logic Colloquium 77. Amsterdam: North-Holland, pp. 55–66.
Beeson, M., 1985. Foundations of constructive mathematics. Metamathematical studies. Ergebnisse der Mathematik under ihrer Grenzgebiete. 3. Folge. Band 6. Berlin: Springer-Verlag. XXIII+466.
Bell, J.L., 1977. Boolean-valued models and independence proofs in set theory. Oxford Logic Guides. Oxford, UK: Clarendon Press. xviii+126.
Diaconescu, R., 1975. Axiom of choice and complementation. Proceedings of the American Mathematical Society, 51, pp. 176–178.
Fraenkel, A., Bar-Hillel, Y., 1958. Foundations of set theory. Studies in logic and the foundations of mathematics. Amsterdam: North-Holland Publishing Company. X+415.
Goodman, N.D., Myhill, J., 1978. Choice implies excluded middle. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 24, p. 461.
Grayson, R.J., 1979. Heyting-valued models for intuitionistic set theory. In: M. Fourman et al. (eds.), Applications of sheaves. Springer Lecture Notes in Mathematics. Volume 753, pp. 402–414.
Henkin, L., 1950. Completeness in the theory of types. The Journal of Symbolic Logic, 15(2), June, pp. 81–91.
Heyting, A., 1956. Intuitionism. An introduction. Amsterdam: North-Holland Publishing Company. IX+137.
Kreisel, G., 1972. Informal rigour and completeness proofs. In: I. Lakatos (ed.), Problems in the philosophy of mathematics. Proceedings of the International Colloquium in the Philosophy of Science, London, 1965, Volume 1. Amsterdam: North-Holland Publishing Company. pp. 138–171.
Monk, J.D., 1969. Introduction to set theory. New York, NY: McGraw-Hill Book Company. ix+193.
McCarty, C., 2018. What is logical truth? Proceedings of the XIV Conference “Dr. Antonio Monteiro.” Universidad Nacional del Sur: Bahía Blanca.
Mycielski, J., Steinhaus, H., 1962. A mathematical axiom contradicting the axiom of choice. Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, 10, pp. 1–3.
Quine, W.V., 1970. Philosophy of logic. Foundations of philosophy series. Englewood Cliffs, NJ: Prentice-Hall, Inc. XIV+109.
Rasiowa, H., Sikorski, R., 1963. The mathematics of metamathematics. Warszawa: Państwowe Wydawnictwo Naukowe. 519 pp.
Scott, D., 1968. Extending the topological interpretation to intuitionistic analysis, I. Compositio Mathematica, 20, pp. 194–210.
Shapiro, S., 1991. Foundations without foundationalism: A case for second-order logic. Oxford Logic Guides. Volume 17. Oxford, UK: Oxford University Press. XX+277.
Tarski, A., 1938. Der Aussagenkalkül und die Topologie. Fundamenta Mathematicae, 31, pp. 103–134.
Troelstra, A., van Dalen, D., 1988. Constructivism in mathematics. Volume I. Studies in Logic and the Foundations of Mathematics. Volume 121. Amsterdam: North-Holland. XX+342+XIV.
Beeson, M., 1985. Foundations of constructive mathematics. Metamathematical studies. Ergebnisse der Mathematik under ihrer Grenzgebiete. 3. Folge. Band 6. Berlin: Springer-Verlag. XXIII+466.
Bell, J.L., 1977. Boolean-valued models and independence proofs in set theory. Oxford Logic Guides. Oxford, UK: Clarendon Press. xviii+126.
Diaconescu, R., 1975. Axiom of choice and complementation. Proceedings of the American Mathematical Society, 51, pp. 176–178.
Fraenkel, A., Bar-Hillel, Y., 1958. Foundations of set theory. Studies in logic and the foundations of mathematics. Amsterdam: North-Holland Publishing Company. X+415.
Goodman, N.D., Myhill, J., 1978. Choice implies excluded middle. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 24, p. 461.
Grayson, R.J., 1979. Heyting-valued models for intuitionistic set theory. In: M. Fourman et al. (eds.), Applications of sheaves. Springer Lecture Notes in Mathematics. Volume 753, pp. 402–414.
Henkin, L., 1950. Completeness in the theory of types. The Journal of Symbolic Logic, 15(2), June, pp. 81–91.
Heyting, A., 1956. Intuitionism. An introduction. Amsterdam: North-Holland Publishing Company. IX+137.
Kreisel, G., 1972. Informal rigour and completeness proofs. In: I. Lakatos (ed.), Problems in the philosophy of mathematics. Proceedings of the International Colloquium in the Philosophy of Science, London, 1965, Volume 1. Amsterdam: North-Holland Publishing Company. pp. 138–171.
Monk, J.D., 1969. Introduction to set theory. New York, NY: McGraw-Hill Book Company. ix+193.
McCarty, C., 2018. What is logical truth? Proceedings of the XIV Conference “Dr. Antonio Monteiro.” Universidad Nacional del Sur: Bahía Blanca.
Mycielski, J., Steinhaus, H., 1962. A mathematical axiom contradicting the axiom of choice. Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques, 10, pp. 1–3.
Quine, W.V., 1970. Philosophy of logic. Foundations of philosophy series. Englewood Cliffs, NJ: Prentice-Hall, Inc. XIV+109.
Rasiowa, H., Sikorski, R., 1963. The mathematics of metamathematics. Warszawa: Państwowe Wydawnictwo Naukowe. 519 pp.
Scott, D., 1968. Extending the topological interpretation to intuitionistic analysis, I. Compositio Mathematica, 20, pp. 194–210.
Shapiro, S., 1991. Foundations without foundationalism: A case for second-order logic. Oxford Logic Guides. Volume 17. Oxford, UK: Oxford University Press. XX+277.
Tarski, A., 1938. Der Aussagenkalkül und die Topologie. Fundamenta Mathematicae, 31, pp. 103–134.
Troelstra, A., van Dalen, D., 1988. Constructivism in mathematics. Volume I. Studies in Logic and the Foundations of Mathematics. Volume 121. Amsterdam: North-Holland. XX+342+XIV.