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Boolean extensions which efface the Mahlo property1

  • William Boos (a1)


The results that follow are intended to be understood as informal counterparts to formal theorems of Zermelo-Fraenkel set theory with choice. Basic notation not explained here can usually be found in [5]. It will also be necessary to assume a knowledge of the fundamentals of boolean and generic extensions, in the style of Jech's monograph [3]. Consistency results will be stated as assertions about the existence of certain complete boolean algebras, B, C , etc., either outright or in the sense of a countable standard transitive model M of ZFC augmented by hypotheses about the existence of various large cardinals. Proofs will usually be phrased in terms of the forcing relation ⊩ over such an M, especially when they make heavy use of genericity. They are then assertions about Shoenfield-style P -generic extensions M(G), in which the ‘names’ are required without loss of generality to be elements of M B = (V B )M, B is the boolean completion of P in M (cf. [3, p. 50]: the notation there is RO(P)), the generic G is named by ĜM B such that (⟦pĜ B = p and (cf. [11, p. 361] and [3, pp. 58–59]), and for pP and c 1, …, cn M B , p ⊩ φ(c 1, …, cn ) iff ⟦φ(c 1, …, cn )⟧ B p (cf. [3, pp. 61–62]).

Some prior acquaintance with large cardinal theory is also needed. At this writing no comprehensive introductory survey is yet in print, though [1], [10], [12]and [13] provide partial coverage. The scheme of definitions which follows is intended to fix notation and serve as a glossary for reference, and it is followed in turn by a description of the results of the paper. We adopt the convention that κ, λ, μ, ν, ρ and σ vary over infinite cardinals, and all other lower case Greek letters (except χ, φ, ψ, ϵ) over arbitrary ordinals.



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This paper is a revision of the first half of the author's Ph.D. thesis, written under the supervision of Professor Kenneth Kunen, and submitted to the University of Wisconsin in August, 1971. I would like to thank Professor Kunen for his persistent encouragement and quietly infallible guidance in teaching me set theory. I would also like to thank Professor Karel Prikry for introducing me to techniques of Jensen and Mitchell that are basic to the results of this paper.



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[1] Chang, C. C. and Keisler, H. J., Model theory, North-Holland, Amsterdam, 1974.
[2] Chang, C. C. et al., Mimeographed Lecture Notes on Set Theory, University of California, Los Angeles, 19671968.
[3] Jech, T., Lectures in set theory, Springer-Verlag, New York, 1971.
[4] Jensen, R. B. and Solovay, R. M., Some applications of almost disjoint sets, Mathematical logic and foundations of set theory (Bar-Hillel, Y., Editor), North-Holland, Amsterdam, 1970.
[5] Krivine, J.-L., Introduction to axiomatic set theory, Reidel, Dordrecht, Holland, and Humanities Press, New York, 1971.
[6] Kunen, K., Some applications of iterated ultrapowers in set theory, Annals of Mathematical Logic, vol. 1 (1970), pp. 179227.
[6a] Kunen, K., Saturated models (to appear).
[7] Kunen, K. and Paris, J. B., Boolean extensions and measurable cardinals, Annals of Mathematical Logic, vol. 2 (1970/1971), pp. 359377.
[8] Mitchell, W., Aronszajn trees and the independence of the transfer property, Annals of Mathematical Logic, vol. 5 (1972), pp. 2146.
[9] Prikry, K., Changing measurable into accessible cardinals, Rozprawy Matematyczne, Warszawa, 1970.
[10] Rowbottom, F. and Bacsich, P. D., Classical theory of large cardinals, Typewritten Lecture Notes, University of California, Los Angeles, 1967.
[11[ Shoenfield, J. R., Unramified forcing, Proceedings of Symposia in Pure Mathematics, vol. 13, part I (Scott, Dana, Editor), American Mathematical Society, Providence, R.I., 1971.
[12] Shoenfield, J. R., Measurable cardinals, Logic Colloquium '69 (Gandy, R. O. and Yates, C. M. E., Editors) North-Holland, Amsterdam, 1971.
[13] Silver, J., Some applications of model theory in set theory, Annals of Mathematical Logic, vol. 3 (1971), pp. 45110.
[14] Solovay, R., Real-valued measurable cardinals, Proceedings of Symposia in Pure Mathematics, vol. 13, part I (Scott, Dana, Editor), American Mathematical Society, Providence, R.I., 1971.


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