Summary
The major point of contention among the philosophers and mathematicians who have written about the independence results for the continuum hypothesis (CH) and related questions in set theory has been the question of whether these results give reason to doubt that the independent statements have definite truth values. This paper concerns the views of G. Kreisel, who gives arguments based on second order logic that the CH does have a truth value. The view defended here is that although Kreisel's conclusion is correct, his arguments are unsatisfactory. Later sections of the paper advance a different argument that the independence results do not show lack of truth values.
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References
J. W. Addison, “Some Consequences of the Axiom of Constructibility”, Fundamenta Mathematicae 46 (1959).
P. Bernays, “What Do Some Recent Results in Set Theory Suggest?”, in Lakatos, The Philosophy of Mathematics, North-Holland Publishing Co., Amsterdam, 1967.
E. Bishop, Foundations of Constructive Analysis, McGraw-Hill Publishers, New Your, 1969.
A. Church, Introduction to Mathematical Logic, Princeton University Press, 1956.
K. Godel, “What is Cantor's Continuum Hypothesis?”, in Benecerraf and Putnam, Philosophy of Mathematics, Prentice-Hall, Englewood Cliffs, 1964.
L. Henkin, “Completeness in the Theory of Types,” Journal of Symbolic Logic 15 (1950).
T. J. Jech, Lectures in Set Theory, Springer-Verlag, New York, 1971.
L. Kalmar, “On the Role of Second Order Theories” in Lakatos, loc. cit. in [2].
G. Kreisel, “Informal Rigour and Completeness Proofs,” in Lakatos, loc. cit. in [2].
G. Kreisel, “Comments on Mostowski's Paper,” in Lakatos, loc. cit. in [2].
G. Kreisel, Appendix II, in S. MacLane, ed., Reports of the Midwest Category Seminar III, Springer-Verlag, New York, 1969.
G. Kreisel, “Observations on Popular Discussions of Foundations,” in D. Scott, Axiomatic Set Theory, American Mathematical Society, Providnece, R.I., 1971.
G. Kreisel, “Two Notes on the Foundations of Set Theory,” Dialectica 23 (1969).
G. Kreisel and J. L. Krivine, Elements of Mathematical Logic (Model Theory), North-Holland Publishing Co., Amsterdam, 1967.
A. Mostowski, “Recent Results in Set Theory,” in Lakatos, loc. cit. in [2].
J. Shepherdson, “Inner Models for Set Theory,” II, Journal of Symbolic Logic 17 (1952).
J. R. Shoenfield, Mathematical Logic, Addison-Wesley Co., Reading, Massachusetts, 1967.
R. Smullyan, “Continuum Hypothesis,” in The Encyclopedia of Philosophy, P. Edwards, ed., Macmillan Co., New York, 1967.
R. Solovay, “A Δ ′3 Nonconstructible Set of Integers,” Transactions of the A.M.S. (1967).
P. Suppes, “After Set Theory, What?”, in Lakatos, loc. cit. in [2].
L. Tharp, Constructibility in Impredicative Set Theory, unpublished Ph.D. dissertation, Massachusetts Institute of Technology, 1965.
A. S. Trolestra, Principles of Intuitionism, Springer-Verlag, New York, 1969.
T. Weston, The Continuum Hypothesis: Independence and Truth Value, unpublished Ph.D. dissertation, Massachusetts Institute of Technology, 1974.
T. Weston, “Theories Whose Quantification Cannot Be Substitutional.” Nous 8 (1974).
E. Zermelo, “Ueber Grenzzahlen und Mengenbereiche,” Fundamenta Mathematicae 16 (1930).
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Weston, T. Kreisel, the continuum hypothesis and second order set theory. J Philos Logic 5, 281–298 (1976). https://doi.org/10.1007/BF00248732
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DOI: https://doi.org/10.1007/BF00248732