Abstract
In [12], Hugh Woodin introduced Ω-logic, an approach to truth in the universe of sets inspired by recent work in large cardinals. Expository accounts of Ω-logic appear in [13, 14, 1, 15, 16, 17]. In this paper we present proofs of some elementary facts about Ω-logic, relative to the published literature, leading up to the generic invariance of Ω-logic and the Ω-conjecture.
Keywords
The first author was partially supported by the research projects BFM2002-03236 of the Ministerio de Ciencia y Tecnología, and 2002SGR 00126 of the Generalitat de Catalunya. The third author was partially supported by NSF Grant DMS-0401603. This paper was written during the third author’s stay at the Centre de Recerca Matemàtica (CRM), whose support under a Mobility Fellowship of the Ministerio de Educación, Cultura y Deportes is gratefully acknowledged. It was finally completed during the first and third authors’ stay at the Institute for Mathematical Sciences, National University of Singapore, in July 2005.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
P. Dehornoy, Progrès récents sur l’hypothèse du continu (d’après Woodin), Séminaire Bourbaki 55ème année, 2002–2003, #915.
Q. Feng, M. Magidor, W.H. Woodin, Universally Baire Sets of Reals. Set Theory of the Continuum (H. Judah, W. Just and W.H. Woodin, eds), MSRI Publications, Berkeley, CA, 1989, pp. 203–242, Springer Verlag 1992.
S. Jackson, Structural consequences of AD, Handbook of Set Theory, M. Foreman, A. Kanamori and M. Magidor, eds. To appear.
T. Jech, Set theory, 3d Edition, Springer, New York, 2003.
A. Kanamori, The Higher Infinite. Large cardinals in set theory from their beginnings. Perspectives in Mathematical Logic. Springer-Verlag. Berlin, 1994.
P.B. Larson, The Stationary Tower. Notes on a course by W. Hugh Woodin. University Lecture Series, Vol. 32. American Mathematical Society, Providence, RI. 2004.
P.B. Larson, Forcing over models of determinacy, Handbook of Set Theory, M. Foreman, A. Kanamori and M. Magidor, eds. To appear.
D.A. Martin, J.R. Steel, The extent of scales in L(ℝ), Cabal seminar 79–81, Lecture Notes in Math. 1019, Springer, Berlin, 1983, 86–96.
Y.N. Moschovakis, Descriptive Set Theory, Studies in Logic and the Foundations of Mathematics. Vol. 100. North-Holland Publishing Company. Amsterdam, New York, Oxford, 1980.
S. Shelah, W.H. Woodin, Large cardinals imply that every reasonably definable set of reals is Lebesgue measurable. Israel J. of Math. vol. 70, n. 3 (1990), 381–394.
J. Steel, A theorem of Woodin on mouse sets. Preprint. July 14, 2004.
W.H. Woodin, The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal. DeGruyter Series in Logic and Its Applications, vol. 1, 1999.
W.H. Woodin, The Continuum Hypothesis. Cori, René (ed.) et al., Logic colloquium 2000. Proceedings of the annual European summer meeting of the Association for Symbolic Logic, Paris, France, July 23–31, 2000. Wellesley, MA: A.K. Peters; Urbana, IL: Association for Symbolic Logic. Lecture Notes in Logic 19, 143–197 (2005).
W.H. Woodin, The Ω-Conjecture. Aspects of Complexity (Kaikoura, 2000). DeGruyter Series in Logic and Its Applications, vol. 4, pages 155–169. DeGruyter, Berlin, 2001.
W.H. Woodin, The Continuum Hypothesis, I. Notices Amer. Math. Soc., 48(6):567–576, 2001.
W.H. Woodin, The Continuum Hypothesis, II. Notices Amer. Math. Soc., 48(7):681–690, 2001; 49(1):46, 2002.
W.H. Woodin, Set theory after Russell; The journey back to Eden. In One Hundred Years of Russell’s Paradox, edited by Godehard Link. DeGruyter Series in Logic and Its Applications, vol. 6, pages 29–48.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Birkhäuser Verlag Basel/Switzerland
About this chapter
Cite this chapter
Bagaria, J., Castells, N., Larson, P. (2006). An Ω-logic Primer. In: Bagaria, J., Todorcevic, S. (eds) Set Theory. Trends in Mathematics. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7692-9_1
Download citation
DOI: https://doi.org/10.1007/3-7643-7692-9_1
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7691-8
Online ISBN: 978-3-7643-7692-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)