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Regularity properties and definability in the real number continuum: idealized forcing, polarized partitions, Hausdorff gaps and mad families in the projective hierarchy

Khomskii, Y.D.

Publication date 2012

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Citation for published version (APA): Khomskii, Y. D. (2012). Regularity properties and definability in the real number continuum: idealized forcing, polarized partitions, Hausdorff gaps and mad families in the projective hierarchy. Institute for Logic, Language and Computation.

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[Abr10] Uri Abraham. Proper forcing. In Matthew Foreman and Akihiro Kanamori, editors, Handbook of Set Theory, pages 333–394. Springer, Dordrecht, 2010.

[Ale16] Pavel S. Aleksandrov. Sur la puissance des ensembles mesurables. C. R. Acad. Sci. Paris, 162:323–325, 1916.

[Bai99] Ren´e-LouisBaire. Sur les fonctions de variables r´eelles. Ann. Mat. Pura Appl., 3(3):1–123, 1899.

[Bar84] Tomek Bartoszy´nski.Additivity of measure implies additivity of cat- egory. Trans. Amer. Math. Soc., 281(1):209–213, 1984.

[BD85] James E. Baumgartner and Peter Dordal. Adjoining dominating func- tions. J. Symbolic Logic, 50(1):94–101, 1985.

[Bel85] John L. Bell. Boolean-valued models and independence proofs in set theory, volume 12 of Oxford Logic Guides. The Clarendon Press Oxford University Press, New York, second edition, 1985. With a foreword by Dana Scott.

[Ber08] Felix Bernstein. Zur Theorie der trigonometrischen Reihen. Ber. Verh. k. S¨achs.Ges. Wiss. Leipzig, Math.-Phys. Kl., 60:325–338, 1908.

[BHL05] J¨orgBrendle, Lorenz Halbeisen, and Benedikt L¨owe. Silver measur- ability and its relation to other regularity properties. Math. Proc. Cambridge Philos. Soc., 138(1):135–149, 2005.

[BHS95] J¨orgBrendle, Greg Hjorth, and Otmar Spinas. Regularity properties for dominating projective sets. Ann. Pure Appl. Logic, 72(3):291–307, 1995.

125 126 Bibliography

[BJ95] Tomek Bartoszy´nskiand Haim Judah. Set Theory, On the Structure of the Real Line. A K Peters, 1995.

[BL99] J¨orgBrendle and Benedikt L¨owe. Solovay-type characterizations for forcing-algebras. J. Symbolic Logic, 64(3):1307–1323, 1999.

[BL11] J¨orgBrendle and Benedikt L¨owe. Eventually different functions and inaccessible cardinals. J. Math. Soc. Japan, 63(1):137–151, 2011.

[Bla10] Andreas Blass. Combinatorial cardinal characteristics of the contin- uum. In Matthew Foreman and Akihiro Kanamori, editors, Handbook of Set Theory, pages 395–489. Springer, Dordrecht, 2010.

[Bre09] J¨orgBrendle. Forcing and the structure of the real line: the bogot´a lectures. Lecture notes, 2009.

[Bre10] J¨orgBrendle. Aspects of iterated forcing. Tutorial at Winter School in Abstract Analysis, section Set Theory & Topology, January/February 2010.

[BY05] J¨orgBrendle and Shunsuke Yatabe. Forcing indestructibility of MAD families. Ann. Pure Appl. Logic, 132(2-3):271–312, 2005.

[Coh63] Paul Cohen. The independence of the continuum hypothesis. Proc. Nat. Acad. Sci. U.S.A., 50:1143–1148, 1963.

[Coh64] Paul J. Cohen. The independence of the continuum hypothesis. II. Proc. Nat. Acad. Sci. U.S.A., 51:105–110, 1964.

[Dev84] Keith J. Devlin. Constructibility. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1984.

[DPLT01] Carlos A. Di Prisco, Jimena Llopis, and Stevo Todorˇcevi´c.Borel parti- tions of products of finite sets and the Ackermann function. J. Combin. Theory Ser. A, 93(2):333–349, 2001.

[DPT03] Carlos A. Di Prisco and Stevo Todorˇcevi´c.Souslin partitions of prod- ucts of finite sets. Adv. Math., 176(1):145–173, 2003.

[Ell74] Erik Ellentuck. A new proof that analytic sets are Ramsey. J. Symbolic Logic, 39:163–165, 1974.

[Fen93] Qi Feng. Homogeneity for open partitions of pairs of reals. Trans. Amer. Math. Soc., 339(2):659–684, 1993.

[FT10] Vera Fischer and Asger T¨ornquist. A co-analytic maximal set of or- thogonal measures. J. Symbolic Logic, 75(4):1403–1414, 2010. Bibliography 127

[FZ10] Sy-David Friedman and Lyubomyr Zdomskyy. Projective mad fami- lies. Ann. Pure Appl. Logic, 161(12):1581–1587, 2010.

[G¨od38] Kurt G¨odel.The consistency of the axiom of choice and of the gener- alized continuum-hypothesis. Proc. Nat. Acad. Sci. USA, 24:556–557, 1938.

[GRSS95] Martin Goldstern, Miroslav Repick´y, Saharon Shelah, and Otmar Spinas. On tree ideals. Proc. Amer. Math. Soc., 123(5):1573–1581, 1995.

[GS53] David Gale and Frank M. Stewart. Infinite games with perfect in- formation. In Contributions to the theory of games, vol. 2, Annals of Mathematics Studies, no. 28, pages 245–266. Princeton University Press, Princeton, N. J., 1953.

[Had94] Jacques Hadamard. Sur les caract`eresde convergence des s´eries`ater- mes positifs et sur les fonctions ind´efiniment croissantes. Acta Math., 18:319–336, 1894.

[Hal03] Lorenz Halbeisen. Making doughnuts of Cohen reals. Math. Logic Q., 49(2):173–178, 2003.

[Hau16] Felix Hausdorff. Die M¨achtigkeit der Borelschen Mengen. Math. Ann., 77(3):430–437, 1916.

[Hau36] Felix Hausdorff. Summen von ℵ1 Mengen. Fund. Math., 26:241–255, 1936.

[HGF03] Michael Hruˇs´akand Salvador Garc´ıaFerreira. Ordering MAD families a la Kat´etov. J. Symbolic Logic, 68(4):1337–1353, 2003.

[HKL90] Leo A. Harrington, Alexander S. Kechris, and Alain Louveau. A Glimm-Effros dichotomy for Borel equivalence relations. J. Amer. Math. Soc., 3(4):903–928, 1990.

[Hru01] Michael Hruˇs´ak. MAD families and the rationals. Comment. Math. Univ. Carolin., 42(2):345–352, 2001.

1 [Iho88] Jaime I. Ihoda. Σ2-sets of reals. J. Symbolic Logic, 53(2):636–642, 1988.

[Ike10a] Daisuke Ikegami. Forcing absoluteness and regularity properties. Ann. Pure Appl. Logic, 161(7):879–894, 2010.

[Ike10b] Daisuke Ikegami. Games in Set Theory and Logic. PhD thesis, Uni- versity of Amsterdam, 2010. ILLC Dissertations DS-2010-04. 128 Bibliography

1 [IS89] Jaime I. Ihoda and Saharon Shelah. ∆2-sets of reals. Ann. Pure Appl. Logic, 42(3):207–223, 1989.

[Jec03] Thomas Jech. Set theory. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded.

[Kan03] Akihiro Kanamori. The higher infinite. Large cardinals in set theory from their beginnings. Springer Monographs in Mathematics. Springer- Verlag, Berlin, second edition, 2003.

[Kas83] Ilias G. Kastanas. On the Ramsey property for sets of reals. J. Sym- bolic Logic, 48(4):1035–1045 (1984), 1983.

[Kec77] Alexander S. Kechris. On a notion of smallness for subsets of the Baire space. Trans. Amer. Math. Soc., 229:191–207, 1977.

[Kec95] Alexander S. Kechris. Classical descriptive set theory, volume 156 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1995.

[Kho09] Yurii Khomskii. A general setting for the pointwise investigation of determinacy. In Logic and its applications, volume 5378 of Lecture Notes in Comput. Sci., pages 185–195. Springer, Berlin, 2009.

[KS09] Jakob Kellner and Saharon Shelah. Decisive creatures and large con- tinuum. J. Symbolic Logic, 74(1):73–104, 2009.

[KSZ08] Bart Kastermans, Juris Stepr¯ans,and Yi Zhang. Analytic and co- analytic families of almost disjoint functions. J. Symbolic Logic, 73(4):1158–1172, 2008.

[Kun80] Kenneth Kunen. Set theory. An introduction to independence proofs, volume 102 of Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam, 1980.

[Kur01] MiloˇsS. Kurili´c.Cohen-stable families of subsets of integers. J. Sym- bolic Logic, 66(1):257–270, 2001.

[Leb02] Henri Lebesgue. Int´egrale,longueur, aire. Ann. Mat. Pura Appl., 3(7):231–359, 1902.

[L¨ow98] Benedikt L¨owe. Uniform unfolding and analytic measurability. Arch. Math. Logic, 37(8):505–520, 1998.

[Luz25] Nikolai N. Luzin. Sur un probl`emede M. Emile Borel et les ensembles projectifs de M. Henri Lebesgue; les ensembles analytiques. C. R. Acad. Sci. Paris, 180:1572–1574, 1925. Bibliography 129

[Man70] Richard Mansfield. Perfect subsets of definable sets of real numbers. Pacific J. Math., 35:451–457, 1970.

[Mar75] Donald A. Martin. Borel determinacy. Ann. of Math. (2), 102(2):363– 371, 1975.

[Mat77] Adrian R. D. Mathias. Happy families. Ann. Math. Logic, 12(1):59– 111, 1977.

[Mil89] Arnold W. Miller. Infinite combinatorics and definability. Ann. Pure Appl. Logic, 41(2):179–203, 1989.

[Mos80] Yiannis N. Moschovakis. Descriptive set theory, volume 100 of Studies in Logic and the Foundations of Mathematics. North-Holland Publish- ing Co., Amsterdam, 1980.

[MS62] Jan Mycielski and H. Steinhaus. A mathematical axiom contradicting the axiom of choice. Bull. Acad. Polon. Sci. S´er.Sci. Math. Astronom. Phys., 10:1–3, 1962.

[MS89] Donald A. Martin and John R. Steel. A proof of projective determi- nacy. J. Amer. Math. Soc., 2(1):71–125, 1989.

[Pri76] Karel Prikry. Determinateness and partitions. Proc. Amer. Math. Soc., 54:303–306, 1976.

[Rag09] Dilip Raghavan. Maximal almost disjoint families of functions. Fund. Math., 204(3):241–282, 2009.

[Ram29] Frank P. Ramsey. On a problem of formal logic. Proc. Lond. Math. Soc. (2), 30:264–286, 1929.

[RS85] Jean Raisonnier and Jacques Stern. The strength of measurability hypotheses. Israel J. Math., 50(4):337–349, 1985.

[RS11] Dilip Raghavan and Saharon Shelah. Comparing the closed almost disjointness and dominating numbers. preprint, 2011.

[Sab10] Marcin Sabok. Complexity of ramsey null sets. preprint, 2010.

[Sch93] Marion Scheepers. Gaps in ωω. In Haim Judah, editor, Set theory of the reals. Proceedings of the Winter Institute held at Bar-Ilan University, Ramat Gan, January 1991, volume 6 of Israel Math. Conf. Proc., pages 439–561. Bar-Ilan Univ., Ramat Gan, 1993.

[She84] Saharon Shelah. Can you take Solovay’s inaccessible away? Israel J. Math., 48(1):1–47, 1984. 130 Bibliography

[Sil70] Jack Silver. Every analytic set is Ramsey. J. Symbolic Logic, 35:60–64, 1970.

[Sol70] Robert M. Solovay. A model of set-theory in which every set of reals is Lebesgue measurable. Ann. of Math. (2), 92:1–56, 1970.

[Spe57] Ernst Specker. Zur Axiomatik der Mengenlehre (Fundierungs- und Auswahlaxiom). Z. Math. Logik Grundlagen Math., 3:173–210, 1957.

[Spi94] Otmar Spinas. Dominating projective sets in the Baire space. Ann. Pure Appl. Logic, 68(3):327–342, 1994.

[Spi04] Otmar Spinas. Analytic countably splitting families. J. Symbolic Logic, 69(1):101–117, 2004.

[Spi08] Otmar Spinas. Perfect set theorems. Fund. Math., 201(2):179–195, 2008.

[Sus17] Mikhail Ya. Suslin. Sur une d´efinitiondes ensembles mesurables B sans nombres transfinis. C. R. Acad. Sci. Paris, 164:88–91, 1917.

[Szp35] Edward Szpilrajn. Sur une classe de fonctions de M. Sierpiski et la classe correspondante d’ensembles. Fund. Math., 24:17–14, 1935.

[Tod89] Stevo Todorˇcevi´c. Partition problems in topology, volume 84 of Con- temporary Mathematics. American Mathematical Society, Providence, RI, 1989.

[Tod96] Stevo Todorˇcevi´c.Analytic gaps. Fund. Math., 150(1):55–66, 1996.

[Tod10] Stevo Todorˇcevi´c. Introduction to Ramsey spaces, volume 174 of An- nals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2010.

[Vit05] Giuseppe Vitali. Sul problema della misura dei gruppi di punti di una retta. Bologna, Gamberini e Parmeggiani, 1905.

[Zap04] Jindˇrich Zapletal. Descriptive set theory and definable forcing, vol- ume 167 of Memoirs of the American Mathematical Society. American Mathematical Society, 2004.

[Zap08] Jindˇrich Zapletal. Forcing idealized, volume 174 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2008.

[ZS10] Jindˇrich Zapletal and Saharon Shelah. Ramsey theorems for products of finite sets with submeasures. preprint; [ShZa:952], 2010.