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Forcing for the impredicative theory of classes

  • Rolando Chuaqui (a1)

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The purpose of this work is to formulate a general theory of forcing with classes and to solve some of the consistency and independence problems for the impredicative theory of classes, that is, the set theory that uses the full schema of class construction, including formulas with quantification over proper classes. This theory is in principle due to A. Morse [9]. The version I am using is based on axioms by A. Tarski and is essentially the same as that presented in [6, pp. 250–281] and [10, pp. 2–11]. For a detailed exposition the reader is referred there. This theory will be referred to as .

The reflection principle (see [8]), valid for other forms of set theory, is not provable in . Some form of the reflection principle is essential for the proofs in the original version of forcing introduced by Cohen [2] and the version introduced by Mostowski [10]. The same seems to be true for the Boolean valued models methods due to Scott and Solovay [12]. The only suitable form of forcing for found in the literature is the version that appears in Shoenfield [14]. I believe Vopěnka's methods [15] would also be applicable. The definition of forcing given in the present paper is basically derived from Shoenfield's definition. Shoenfield, however, worked in Zermelo-Fraenkel set theory.

I do not know of any proof of the consistency of the continuum hypothesis with assuming only that is consistent. However, if one assumes the existence of an inaccessible cardinal, it is easy to extend Gödel's consistency proof [4] of the axiom of constructibility to .

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References

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[1]Bernays, P., What do some recent results in set theory suggest?, Problems in the philosophy of mathematics (Latakos, I., Editor), Studies in logic, North-Holland, Amsterdam, 1967, pp. 109112.
[2]Cohen, P. J., Set theory and the continuum hypothesis, Benjamin, New York, 1966.
[3]Easton, W., Powers of regular cardinals, Thesis, Princeton University, Princeton, N.J., 1964.
[4]Gödel, K., The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory, Annals of Mathematics Studies, No. 3, Princeton Univ. Press, Princeton, N.J., 1940.
[5]Jensen, R., The generalized continuum hypothesis and measurable cardinals, Proceeding of Symposia in Pure Mathematics, vol. 13, American Mathematical Society, Providence, R.I. (to appear).
[6]Kelly, J. L., General topology, Van Nostrand, Princeton, N J., 1955.
[7]Kreisel, G., Informal rigour and completeness proofs, Problems in the philosophy of mathematics (Lakatos, I., Editor), Studies in logic, North-Holland, Amsterdam, 1967, pp. 138171.
[8]Lévy, A., Axiom schemata of strong infinity in axiomatic set theory, Pacific journal of mathematics, vol. 10 (1960), pp. 223238.
[9]Morse, A., A theory of sets, Academic Press, New York, 1965.
[10]Mostowski, A., Constructible sets with applications, Studies in logic, North-Holland, Amsterdam, 1969.
[11]Mostowski, A., An undecidable arithmetical statement, Fundamenta mathematicae, vol. 36 (1949), pp. 143164.
[12]Scott, D. and Solovay, K., Boolean-valued models for set theory, Proceedings of Symposia in Pure Mathematics, vol. 13, American Mathematical Society, Providence, R.I. (to appear).
[13]Shepherdson, J. C., Inner models for set theory. I, this Journal, vol. 16 (1951), pp. 161190.
[14]Shoenfield, J., Unramified forcing, Proceedings of Symposia in Pure Mathematics, vol. 13, Part I, American Mathematical Society, Providence, R.I., 1971, pp. 357382.
[15]Vopěnka, P., The limits of sheaves and applications on constructions of models, Bulletin de l'académie des sciences et des arts. Série des sciences mathématiques, astronomiques et physiques, vol. 13 (1965), pp. 189192.

Forcing for the impredicative theory of classes

  • Rolando Chuaqui (a1)

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