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Sacks forcing, Laver forcing, and Martin’s axiom. (English) Zbl 0755.03026
This paper is mainly about certain cardinals associated with Marczewski’s ideal \(s^ 0\). The cardinals considered here include \(\text{add}(s^ 0)\) and \(\text{cov}(s^ 0)\), which denote the least cardinal \(\kappa\) such that the ideal \(s^ 0\) is not \(\kappa^ +\)-additive and the least \(\kappa\) such that \(2^ \omega\) can be covered by \(\kappa\) sets in \(s^ 0\), respectively. Some of the theorems proved about these cardinals are as follows: (a) \(\text{MA}+\neg\text{CH}\) implies that \(\text{add}(s^ 0)\) is the least cardinal \(\kappa\) such that \(p\Vdash_{\text{Sacks forcing}}\) “\(\text{cof}(\mathfrak c)=\kappa\)” for some \(p\); (b) If one adds \(\omega_ 2\) Sacks reals iteratively with countable supports to a model of CH, then \(\text{add}(s^ 0)=\omega_ 1\) and \({\mathfrak c}=\omega_ 2=\text{cov}(s^ 0)\) in the extension; (c) There is a model of \(\text{MA}+\neg\text{CH}\) in which \(\text{add}(s^ 0)=\text{cov}(s^ 0)=\omega_ 1\). Theorem (c) has also been obtained independently by B. Velickovic [Compos. Math. 79, 279-294 (1991; Zbl 0735.03023)]. It follows from (a) and (c) that it is consistent with \(\text{MA}+\neg\text{CH}\) for Sacks forcing to collapse cardinals. By way of contrast, the authors include the theorem that MA implies that Laver forcing preserves all cardinals. There is an appendix in which it is shown that if \({\mathfrak c}=\omega_ 2\), then Sacks forcing adds a stationary subset of \(\omega_ 2\) that does not have a ground model stationary subset.
Reviewer: J.Takahashi (Kobe)

MSC:
03E35 Consistency and independence results
03E40 Other aspects of forcing and Boolean-valued models
03E50 Continuum hypothesis and Martin’s axiom
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[1] Abraham, U.: A minimal model for ?CH: iteration of Jensen’s reals. Trans. Am. Math. Soc.281, 657-674 (1984) · Zbl 0541.03029
[2] Abraham, U., Rubin, M., Shelah, S.: On the consistency of some partition theorems for continuous colorings and the structure of ?1 real order types. Ann. Pure and Appl. Logic29, 123-206 (1985) · Zbl 0585.03019 · doi:10.1016/0168-0072(84)90024-1
[3] Abraham, U., Toodorcevic, S.: Martin’s axiom and first-countable S- and L-spaces. In: Handbook of set theoretic topology. Kunen, K., Vaughan, J.E. (eds.) North-Holland 1984, pp. 327-346
[4] Aniszczyk, B., Frankiewicz, R., Plewik, S.: Remarks on (s) and Ramsey-measurable functions. Bull. Pol. Acad. of Sci. Math.35, 479-485 (1987) · Zbl 0662.28002
[5] Aniszczyk, B.: Remarks on ?-algebra of (s)-measurable sets. Bull. Pol. Acad. Sci. Math.35, 561-563 (1987) · Zbl 0648.28001
[6] Balcar, B., Vojtas, P.: Refining systems on boolean algebras. (Lect. Notes Math., vol. 619, pp. 45-68) Berlin Heidelberg New York: Springer 1977 · Zbl 0393.06006
[7] Baumgartner, J.: Iterated forcing. In: Mathias, A.R.D. (eds.) Surveys in set theory. London Mathematical Society (Lect. Notes Math., vol. 87, pp. 1-59) Cambridge University Press, Cambridge 1983
[8] Baumgartner, J., Laver, R.: Iterated perfect set forcing. Ann. Math. Logic17, 271-288 (1979) · Zbl 0427.03043 · doi:10.1016/0003-4843(79)90010-X
[9] Blass, A., Shelah, S.: Near coherence of filters. III. A simplified consistency proof. Notre Dame J. Formal Logic30, 530-538 (1989) · Zbl 0702.03030 · doi:10.1305/ndjfl/1093635236
[10] Blass, A.: Applications of superperfect forcing and its relatives. In: Steprans, J., Watson, S. (eds.) Set theory and its applications. York University 1987 (Lect. Notes Math., vol. 1401, pp. 18-40) Berlin Heidelberg New York: Springer 1989
[11] Brown, J.B.: Singular sets and Baire order. Real Anal. Exch.10, 78-84 (1985) · Zbl 0579.54021
[12] Brown, J.B.: Marczewski null sets and intermediate Baire order. Contemp. Math.94, 43-50 (1989) · Zbl 0711.28001
[13] Brown, J.B.: The Ramsey sets and related sigma algebras and ideals. (Preprint) · Zbl 0737.28004
[14] Brown, J.B., Cox, G.V.: Classical theory of totally imperfect sets. Real Anal. Exch.7, 185-232 (1982) · Zbl 0503.54045
[15] Brown, J.B., Prikry, K.: Variations on Lusin’s theorem. Trans. Am. Math. Soc.302, 77-86 (1987) · Zbl 0619.28005
[16] Corazza, P.: The generalized Borel conjecture and strongly proper orders. Trans. Am. Math. Soc.316, 115-140 (1989) · Zbl 0693.03031 · doi:10.1090/S0002-9947-1989-0982239-X
[17] Devlin, K.: ?1-trees. Ann. Math. Logic13, 267-330 (1978) · Zbl 0397.03035
[18] Gurevich, Y., Shelah, S.: Monadic theory of order and topology in ZFC. Ann. Math. Logic23, 179-198 (1982) · Zbl 0516.03007 · doi:10.1016/0003-4843(82)90004-3
[19] Judah, H., Shelah, S.: The Kunen-Miller chart. J. Symb. Logic55, 909-927 (1990) · Zbl 0718.03037 · doi:10.2307/2274464
[20] Kechris, A.: On a notion of smallness for subsets of the Baire space. Trans. Am. Math. Soc.229, 191-207 (1977) · Zbl 0401.03022 · doi:10.1090/S0002-9947-1977-0450070-1
[21] Kunen, K.: Set theory. Amsterdam: North-Holland 1980 · Zbl 0443.03021
[22] Kunen, K.: Where MA first fails. J. Symb. Logic53, 429-433 (1988) · Zbl 0681.03036 · doi:10.2307/2274515
[23] Laver, R.: On the consistency of Borel’s conjecture. Acta Math.137, 151-169 (1976) · Zbl 0357.28003 · doi:10.1007/BF02392416
[24] Marczewski, E.: Sur une classe de fonctions de W. Sierpi?ski et la classe correspondante d’ensembles. Fundam. Math.24, 17-34 (1935) · JFM 61.0229.01
[25] Matet. P.: Personal communication
[26] Mathias, A.R.D.: Happy Families. Ann. Math. Logic12, 59-111 (1977) · Zbl 0369.02041 · doi:10.1016/0003-4843(77)90006-7
[27] Miller, A.: Rational perfect set forcing. Contemp. Math. (American Mathematical Society)31, 143-159 (1984) · Zbl 0555.03020
[28] Miller, A.: Special subsets of the real line. In: Kunen, K., Vaughan, J. (eds.) Handbook of set theoretic topology. Amsterdam: North-Holland 1984, pp. 201-234
[29] Morgan II, J.C.: On the general theory of point sets. II. Real Anal. Exch.12, 377-386 (1986) · Zbl 0629.28001
[30] Mycielski, J.: Some new ideals of subsets on the real line. Colloq. Math.20, 71-76 (1969) · Zbl 0203.05701
[31] Pawlikowski, J.: Parametrized Ellentuck theorem. Topology Appl.37, 65-73 (1990) · Zbl 0734.04001 · doi:10.1016/0166-8641(90)90015-T
[32] Sacks, G.E.: Forcing with perfect closed sets, axiomatic set theory. Proc. Symp. Pure Math.13, 331-355 (1971)
[33] Shelah, S.: A weak generalization of MA to higher cardinals. Isr. J. Math.30, 297-306 (1978) · Zbl 0384.03032 · doi:10.1007/BF02761994
[34] Silver, J.: Every analytic set is Ramsey. J. Symb. Logic.35, 60-64 (1970) · Zbl 0216.01304 · doi:10.2307/2271156
[35] Steprans, J.: The covering number of the Mycielski ideal. (Preprint) · Zbl 0916.03033
[36] Velickovic, B.: CCC posets of perfect trees. (Preprint) · Zbl 0735.03023
[37] Weiss, W.: Versions of Martin’s Axiom. In: Kunen, K., Vaughan, J.E. (eds.) Handbook of set theoretic topology. Amsterdam: North-Holland 1984, pp. 827-886
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