SOME SET THEORIES ARE MORE EQUAL ... - Logic at Harvard

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24 MENACHEM MAGIDORSo not all set theories are equal . In Orwellian language ”Some settheories are more equal”. The challenge to the set theorists is to makesure that ”The set Set Theory will win!”.References[1] Uri Abraham. Proper forcing. In Handbook of set theory. Vols. 1, 2, 3, pages333–394. Springer, Dordrecht, 2010.[2] Uri Abraham and Saharon Shelah. A ∆ 2 2 well-order of the reals and incompactnessof L(Q MM ). Ann. Pure Appl. Logic, 59(1):1–32, 1993.[3] Uri Abraham and Saharon Shelah. Coding with ladders a well ordering of thereals. J. Symbolic Logic, 67(2):579–597, 2002.[4] J. S. Bell. On the einstein podolsky-rosen paradox. Physics, 1:195–200, 1964.[5] James Cummings and Menachem Magidor. Martin’s maximum and weaksquare. Proc. Amer. Math. Soc., 139(9):3339–3348, 2011.[6] Ilijas Farah, Richard Ketchersid, Paul Larson, and Menachem Magidor. Absolutenessfor universally Baire sets and the uncountable. II. In Computationalprospects of infinity. Part II. Presented talks, volume 15 of Lect. Notes Ser.Inst. Math. Sci. Natl. Univ. Singap., pages 163–191. World Sci. Publ., Hackensack,NJ, 2008.[7] Ilijas Farah and Menachem Magidor. Pitowsky functions. preprint.[8] Solomon Feferman. Is the continuum hypothesis a definite mathematical problem?A paper submitted for the lecture in the EFI project, 2011.[9] Solomon Feferman, Harvey M. Friedman, Penelope Maddy, and John R. Steel.Does mathematics need new axioms? Bull. Symbolic Logic, 6(4):401–446, 2000.[10] Qi Feng, Menachem Magidor, and Hugh Woodin. Universally Baire sets ofreals. In Set theory of the continuum (Berkeley, CA, 1989), volume 26 of Math.Sci. Res. Inst. Publ., pages 203–242. Springer, New York, 1992.[11] M. Foreman, M. Magidor, and S. Shelah. Martin’s maximum, saturated ideals,and nonregular ultrafilters. I. Ann. of Math. (2), 127(1):1–47, 1988.[12] Matthew Foreman and Menachem Magidor. Large cardinals and definablecounterexamples to the continuum hypothesis. Ann. Pure Appl. Logic,76(1):47–97, 1995.[13] Matthew D. Foreman. Has the continuum hypothesis been settled?www.math.helsinki.fi/logic/LC2003/presentations/foreman.pdf, 2006.[14] Moti Gitik. A certain generalization of spfa to higher cardinals. Unpublishedpreprint.[15] Kurt Gödel. What is Cantor’s continuum problem? Amer. Math. Monthly,54:515–525, 1947.[16] Kurt Gödel. Collected works. Vol. III. The Clarendon Press Oxford UniversityPress, New York, 1995. Unpublished essays and lectures, With a prefaceby Solomon Feferman, Edited by Feferman, John W. Dawson, Jr., WarrenGoldfarb, Charles Parsons and Robert M. Solovay.[17] Joel David Hamkins. The set-theoretic multiverse. ArXiv:1108.4223v1[math.LO].[18] Leo Harrington. Long projective wellorderings. Ann. Math. Logic, 12(1):1–24,1977.
SOME SET THEORIES ARE MORE EQUAL 25[19] Simon Kochen and E. P. Specker. The problem of hidden variables in quantummechanics. J. Math. Mech., 17:59–87, 1967.[20] Peter Koellner. On reflection principles. Ann. Pure Appl. Logic, 157(2-3):206–219, 2009.[21] Peter Koellner. The continuum hypothesis. General Backgroung Material forthe EFI project,logic.harvard.edu/EFI CH.pdf, 2011.[22] Paul B. Larson. Martin’s maximum and definability in H(ℵ 2 ). Ann. Pure Appl.Logic, 156(1):110–122, 2008.[23] Penelope Maddy. Naturalism in mathematics. The Clarendon Press OxfordUniversity Press, New York, 1997.[24] Penelope Maddy. How applied mathematics became pure. Review of SymbolicLogic, 1:16–41, 2008.[25] Penelope Maddy. Defending the axioms: on the philosophical foundations ofset theory. Oxford University Press, Oxford, 2011.[26] D. A. Martin and R. M. Solovay. Internal Cohen extensions. Ann. Math. Logic,2(2):143–178, 1970.[27] Donald A. Martin. Hilbert’s first problem: the continuum hypothesis. InMathematical developments arising from Hilbert problems (Proc. Sympos. PureMath., Northern Illinois Univ., De Kalb, Ill., 1974), pages 81–92. Amer. Math.Soc., Providence, R. I., 1976.[28] Justin Tatch Moore. Set mapping reflection. J. Math. Log., 5(1):87–97, 2005.[29] Andrzej Mostowski. On some new metamathematical results concerning settheory. Studia Logica, 20:99–116, 1967.[30] Itay Neeman. A high analog of the proper forcing axiom. persoonal communication.[31] Itay Neeman. Optimal proofs of determinacy. Bull. Symbolic Logic, 1(3):327–339, 1995.[32] Itamar Pitowsky. Resolution of the Einstein-Podolsky-Rosen and Bell paradoxes.Phys. Rev. Lett., 48(19):1299–1302, 1982.[33] Itamar Pitowsky. Deterministic model of spin and statistics. Phys. Rev. D (3),27(10):2316–2326, 1983.[34] Itamar Pitowsky. Quantum mechanics and value definiteness. Philos. Sci.,52(1):154–156, 1985.[35] Ernest Schimmerling. Coherent sequences and threads. Adv. Math., 216(1):89–117, 2007.[36] Stewart Shapiro. Thinking about mathematics. Oxford University Press, Oxford,2000. The philosophy of mathematics.[37] Saharon Shelah. Proper and improper forcing. Perspectives in MathematicalLogic. Springer-Verlag, Berlin, second edition, 1998.[38] Abner Shimony. Bellś theorem. In Edward N. Zalta, editor, The Stanford Encyclopediaof Philosophy. Summer 2009 edition, 2009.[39] Joseph Shipman. Cardinal conditions for strong Fubini theorems. Trans. Amer.Math. Soc., 321(2):465–481, 1990.[40] Jonathan Stavi and Jouko Väänänen. Reflection principles for the continuum.In Logic and algebra, volume 302 of Contemp. Math., pages 59–84. Amer. Math.Soc., Providence, RI, 2002.[41] Boban Veličković. Forcing axioms and stationary sets. Adv. Math., 94(2):256–284, 1992.
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