Abstract
This chapter provides an introduction to the basic theory of inner models of set theory, without fine structure. Section 1 begins with the basic theory of Gödel’s class L of constructible sets, with an emphasis on the condensation property, introduces sharps, and includes a brief discussion of the Dodd-Jensen core model. The next two sections describe the extension of these concepts to arbitrary sequences of measures, and then via extender models to cardinal properties stronger than measurability. Section 4 gives a summary of the status and known properties of inner models for cardinals ranging from strong to supercompact, and the final section discusses core models.
This research was partly supported by NSF grant DPS-9970536. The author would like to thank John Krueger, Paul Larson, Diego Rojas, and the referee for valuable corrections and suggestions.
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Bibliography
Stewart Baldwin. Between strong and superstrong. The Journal of Symbolic Logic, 51(3):547–559, 1986.
Aaron Beller, Ronald B. Jensen, and Philip D. Welch. Coding the Universe, volume 47 of London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, 1982.
Keith J. Devlin. Constructibility. Springer, Berlin, 1984.
Anthony J. Dodd. Core models. The Journal of Symbolic Logic, 48(1):78–90, 1983.
Anthony J. Dodd and Ronald B. Jensen. The core model. Annals of Mathematical Logic, 20(1):43–75, 1981.
Anthony J. Dodd and Ronald B. Jensen. The covering lemma for K. Annals of Mathematical Logic, 22(1):1–30, 1982.
Anthony J. Dodd and Ronald B. Jensen. The covering lemma for L[U]. Annals of Mathematical Logic, 22(2):127–135, 1982.
Matthew Foreman, Menachem Magidor, and Saharon Shelah. Martin’s maximum, saturated ideals and nonregular ultrafilters. II. Annals of Mathematics (2), 127(3):521–545, 1988.
Matthew Foreman, Menachem Magidor, and Saharon Shelah. Martin’s maximum, saturated ideals, and nonregular ultrafilters. I. Annals of Mathematics (2), 127(1):1–47, 1988.
Sy D. Friedman. Constructibility and class forcing. Chapter 8 in this Handbook. 10.1007/978-1-4020-5764-9_9.
Sy D. Friedman. Provable \({\Pi\sp1\sb2}\) -singletons. Proceedings of the American Mathematical Society, 123(9):2873–2874, 1995.
Moti Gitik. Prikry-type forcings. Chapter 16 in this Handbook. 10.1007/978-1-4020-5764-9_17.
Moti Gitik. On closed unbounded sets consisting of former regulars. The Journal of Symbolic Logic, 64(1):1–12, 1999.
Thomas J. Jech. Set Theory. Springer Monographs in Mathematics. Springer, Berlin, 2002. Third millennium edition, revised and expanded.
Akihiro Kanamori. The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings. Springer, Berlin, 1994.
Akihiro Kanamori. The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings. Springer, Berlin, 2003. Second edition.
Peter Koellner and W. Hugh Woodin. Large cardinals from determinacy. Chapter 23 in this Handbook. 10.1007/978-1-4020-5764-9_24.
Kenneth Kunen. Some applications of iterated ultrapowers in set theory. Annals of Mathematical Logic, 1:179–227, 1970.
Kenneth Kunen. Elementary embeddings and infinitary combinatorics. The Journal of Symbolic Logic, 36:407–413, 1971.
Paul B. Larson. The Stationary Tower. Notes on a Course by W. Hugh Woodin, volume 32 of University Lecture Series. American Mathematical Society, Providence, 2004.
Menachem Magidor. Changing cofinality of cardinals. Fundamenta Mathematicae, 99(1):61–71, 1978.
Donald A. Martin and John R. Steel. Iteration trees. Journal of the American Mathematical Society, 7(1):1–73, 1994.
Adrian R. D. Mathias. On sequences generic in the sense of Prikry. Journal of the Australian Mathematical Society, 15:409–414, 1973.
William J. Mitchell. The covering lemma. Chapter 18 in this Handbook. 10.1007/978-1-4020-5764-9_19.
William J. Mitchell. Sets constructible from sequences of ultrafilters. The Journal of Symbolic Logic, 39:57–66, 1974.
William J. Mitchell. How weak is a closed unbounded ultrafilter? In Dirk van Dalen, Daniel Lascar, and Timothy J. Smiley, editors, Logic Colloquium ’80 (Prague, 1998), pages 209–230. North-Holland, Amsterdam, 1982.
William J. Mitchell. The core model up to a Woodin cardinal. In Dag Prawitz, Brian Skyrms, Dag Westerstahl, editors, Logic, Methodology and Philosophy of Science, IX (Uppsala, 1991), volume 134 of Studies in Logic and the Foundations of Mathematics, pages 157–175. North-Holland, Amsterdam, 1994.
William J. Mitchell. A measurable cardinal with a closed unbounded set of inaccessibles from o(κ)=κ. Transactions of the American Mathematical Society, 353(12):4863–4897, 2001.
Itay Neeman. Determinacy in L(ℝ). Chapter 22 in this Handbook. 10.1007/978-1-4020-5764-9_23.
Itay Neeman. Inner models in the region of a Woodin limit of Woodin cardinals. Annals of Pure and Applied Logic, 116(1–3):67–155, 2002.
Karl Prikry. Changing measurable into accessible cardinals. Dissertationes Mathematicae (Rozprawy Mathematycne), 68:359–378, 1971.
Lon B. Radin. Adding closed cofinal sequences to large cardinals. Annals of Mathematical Logic, 22(3):243–261, 1982.
Ernest Schimmerling. A core model toolbox and guide. Chapter 20 in this Handbook. 10.1007/978-1-4020-5764-9_21.
Ernest Schimmerling and Martin Zeman. Characterization of \(\square\sb\kappa\) in core models. Journal of Mathematical Logic, 4(1):1–72, 2004.
Ralf Schindler. The core model for almost linear iterations. Annals of Pure and Applied Logic, 116(1–3):205–272, 2002.
Ralf Schindler and Martin Zeman. Fine structure. Chapter 9 in this Handbook. 10.1007/978-1-4020-5764-9_10.
Dana S. Scott. Measurable cardinals and constructible sets. Bulletin de l’Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques, 9:521–524, 1955.
John R. Steel. An outline of inner model theory. Chapter 19 in this Handbook. 10.1007/978-1-4020-5764-9_20.
John R. Steel. Core models with more Woodin cardinals. The Journal of Symbolic Logic, 67(3):1197–1226, 2002.
Philip D. Welch. Σ* fine structure. Chapter 10 this Handbook. 10.1007/978-1-4020-5764-9_11.
W. Hugh Woodin. The Axiom of Determinacy, Forcing Axioms, and the Nonstationary Ideal. de Gruyter, Berlin, 1999.
Martin Zeman. Inner Models and Large Cardinals. de Gruyter, Berlin, 2002.
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Mitchell, W.J. (2010). Beginning Inner Model Theory. In: Foreman, M., Kanamori, A. (eds) Handbook of Set Theory. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5764-9_18
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