Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-22T14:16:12.808Z Has data issue: false hasContentIssue false
  • English
  • Français

Descriptive inner model theory

Published online by Cambridge University Press:  05 September 2014

Grigor Sargsyan
Grigor Sargsyan*
Affiliation:
Department of Mathematics, Rutgers University, Hill Center for the Mathematical Sciences, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA E-mail: [email protected] URL: http://math.rutgers.edu/~gs481
Get access Rights & Permissions [Opens in a new window]

Abstract

The purpose of this paper is to outline some recent progress in descriptive inner model theory, a branch of set theory which studies descriptive set theoretic and inner model theoretic objects using tools from both areas. There are several interlaced problems that lie on the border of these two areas of set theory, but one that has been rather central for almost two decades is the conjecture known as the Mouse Set Conjecture (MSC). One particular motivation for resolving MSC is that it provides grounds for solving the inner model problem which dates back to 1960s. There have been some new partial results on MSC and the methods used to prove the new instances suggest a general program for solving the full conjecture. It is then our goal to communicate the ideas of this program to the community at large.

Keywords

03E1503E4503E60Mouseinner model theorydescriptive set theoryhod mouse
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 19 , Issue 1 , March 2013 , pp. 1 - 55
Copyright
Copyright © Association for Symbolic Logic 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Abraham, Uri and Shelah, Saharon, Coding with ladders a well ordering of the reals, The Journal of Symbolic Logic, vol. 67 (2002), no. 2, pp. 579–597.CrossRefGoogle Scholar
[2] Becker, Howard S. and Moschovakis, Yiannis N., Measurable cardinals in playful models, Cabal Seminar 77–79, Lecture Notes in Mathematics, vol. 839, Springer, Berlin, 1981, pp. 203–214.CrossRefGoogle Scholar
[3] Caicedo, Andres, Larson, Paul, Sargsyan, Grigor, Schindler, Ralf, Steel, John R., and Zeman, Martin, Square in ℙmax extensions, in preparation.Google Scholar
[4] Cummings, James, Strong ultrapowers and long core models, The Journal of Symbolic Logic, vol. 58 (1993), no. 1, pp. 240–248.CrossRefGoogle Scholar
[5] Farah, Ilijas, The extender algebra and Σ1 2 absoluteness, The Cabal Seminar (Kechris, Alexander S., Löwe, Benedikt, and Steel, John R., editors), vol. IV, to appear.Google Scholar
[6] Feferman, Solomon, Friedman, Harvey M., Maddy, Penelope, and Steel, John R., Does mathematics need new axioms?, this Bulletin, vol. 6 (2000), no. 4, pp. 401–446.Google Scholar
[7] Feng, Qi, Magidor, Menachem, and Woodin, Hugh, Universally Baire sets of reals, Set theory of the continuum (Berkeley, CA, 1989), Mathematical Sciences Research Institute Publications, vol. 26, Springer, 1992, pp. 203–242.CrossRefGoogle Scholar
[8] Foreman, M., Magidor, M., and Shelah, S., Martin's maximum, saturated ideals, and nonregular ultrafilters. I, Annals of Mathematics. Second Series, vol. 127 (1988), no. 1, pp. 1–47.CrossRefGoogle Scholar
[9] Jackson, Steve, Structural consequences of AD, Handbook of set theory (Foreman, Matthew and Kanamori, Akihiro, editors), vol. 3, Springer, 2010, pp. 1753–1876.CrossRefGoogle Scholar
[10] Jech, Thomas, Set theory, the third millennium ed., Springer Monographs in Mathematics, Springer, Berlin, 2003.Google Scholar
[11] Jensen, Ronald, The fine structure of the constructible hierarchy, Annals of Pure and Applied Logic, vol. 4 (1972), pp. 229–308; erratum, ibid. 4 (1972), 443, with a section by Jack Silver.Google Scholar
[12] Jensen, Ronald, Inner models and large cardinals, this Bulletin, vol. 1 (1995), no. 4, pp. 393–407.Google Scholar
[13] Jensen, Ronald, Schimmerling, Ernest, Schindler, Ralf, and Steel, John, Stacking mice, The Journal of Symbolic Logic, vol. 74 (2009), no. 1, pp. 315–335.CrossRefGoogle Scholar
[14] Kanamori, Akihiro, The higher infinite, Perspectives in Mathematical Logic, Springer, Berlin, 1994.Google Scholar
[15] Kechris, Alexander S., Kleinberg, Eugene M., Moschovakis, Yiannis N., and Woodin, W. Hugh, The axiom of determinacy, strong partition properties and nonsingular measures, Cabal Seminar 77–79, Lecture Notes in Mathematics, vol. 839, Springer, Berlin, 1981, pp. 75–99.CrossRefGoogle Scholar
[16] Koellner, Peter and Woodin, W. Hugh, Large cardinals from determinacy, Handbook of set theory (Foreman, Matthew and Kanamori, Akihiro, editors), vol. 3. Springer, Dordrecht, 2010, pp. 1951–2119.CrossRefGoogle Scholar
[17] Larson, Paul B., The stationary tower, University Lecture Series, vol. 32, American Mathematical Society, Providence, RI, 2004, notes on a course by W. Hugh Woodin.Google Scholar
[18] Maddy, Penelope, Believing the axioms. I, The Journal of Symbolic Logic, vol. 53 (1988), no. 2, pp. 481–511.CrossRefGoogle Scholar
[19] Maddy, Penelope, Believing the axioms. II, The Journal of Symbolic Logic, vol. 53 (1988). no. 3. pp. 736–764.CrossRefGoogle Scholar
[20] Martin, Donald A. and Steel, John R., The extent of scales in L(R). Cabal seminar 79–81, Lecture Notes in Mathematics, vol. 1019, Springer, Berlin, 1983, pp. 86–96.CrossRefGoogle Scholar
[21] Martin, Donald A. and Steel, John R., Iteration trees, Journal of the American Mathematical Society, vol. 7 (1994), no. 1, pp. 1–73.CrossRefGoogle Scholar
[22] Mitchell, William J., Sets constructible from sequences of ultrafilters. The Journal of Symbolic Logic, vol. 39 (1974), pp. 57–66.CrossRefGoogle Scholar
[23] Mitchell, William J., Sets constructed from sequences of measures: revisited, The Journal of Symbolic Logic, vol. 48 (1983), no. 3, pp. 600–609.CrossRefGoogle Scholar
[24] Mitchell, William J., Inner models for large cardinals, available at http://www.math.ufl.edu/~wjm/.Google Scholar
[25] Mitchell, William J. and Steel, John R.. Fine structure and iteration trees. Lecture Notes in Logic, vol. 3, Springer, Berlin, 1994.CrossRefGoogle Scholar
[26] Moschovakis, Yiannis N., Descriptive set theory. Studies in Logic and the Foundations of Mathematics, vol. 100, North-Holland, Amsterdam, 1980.Google Scholar
[27] Neeman, Itay, Optimal proofs of determinacy. this Bulletin, vol. 1 (1995), no. 3, pp. 327–339.Google Scholar
[28] Neeman, Itay, Inner models in the region of a Woodin limit of Woodin cardinals. Annals of Pure and Applied Logic, vol. 116 (2002), no. 1–3, pp. 67–155.CrossRefGoogle Scholar
[29] Neeman, Itay, The determinacy of long games, de Gruyter Series in Logic and its Applications, vol. 7, Walter de Gruyter, Berlin, 2004.CrossRefGoogle Scholar
[30] Rudominer, Mitch, Mouse sets. Annals of Pure and Applied Logic, vol. 87 (1997). no. 1, p. 100.CrossRefGoogle Scholar
[31] Rudominer, Mitch, The largest countable inductive set is a mouse set, The Journal of Symbolic Logic, vol. 64 (1999), no. 2, pp. 443–459.CrossRefGoogle Scholar
[32] Rudominer, Mitchell, Mouse sets definable in L(ℝ), Ph.D. thesis. University of California, Los Angeles, Ann Arbor, MI, 1995.Google Scholar
[33] Sacks, Gerald E., Higher recursion theory, Perspectives in Mathematical Logic. Springer, Berlin, 1990.CrossRefGoogle Scholar
[34] Sargsyan, Grigor, A tale of hybrid mice, Thesis, UC Berkeley. 05 2009.Google Scholar
[35] Sargsyan, Grigor, On the strength of PFA I, available at http://math.rutgers.edu/~gs481/.Google Scholar
[36] Sargsyan, Grigor, A tale of hybrid mice, available at http://math.rutgers.edu/~gs481/.Google Scholar
[37] Sargsyan, Grigor and Zhu, Yizheng, Equiconsistency for AD + “Θ is regidar”, in preparation.Google Scholar
[38] Schimmerling, Ernest, The ABC's of mice, this Bulletin, vol. 7 (2001), no. 4, pp. 485–503.Google Scholar
[39] Schimmerling, Ernest, A core model toolbox and guide. Handbook of set theory (Foreman, Matthew and Kanamori, Akihiro, editors), vol. 3, Springer, Dordrecht, 2010, pp. 1685–1751.CrossRefGoogle Scholar
[40] Schindler, Ralf and Steel, John R., The core model induction, available at http://math.berkeley.edu/~steel.Google Scholar
[41] Solovay, Robert M., The independence of DC from AD, Cabal Seminar 76–77, Lecture Notes in Mathematics, vol. 689, Springer, Berlin, 1978, pp. 171–183.CrossRefGoogle Scholar
[42] Steel, John R.. A theorem of Woodin on mouse sets, available at http://math.berkeley.edu/~steel/.Google Scholar
[43] Steel, John R., Scales in L(ℝ), Cabal seminar 79–81, Lecture Notes in Mathematics, vol. 1019, Springer, Berlin, 1983, pp. 107–156.CrossRefGoogle Scholar
[44] Steel, John R., HODL(ℝ) is a core model below Θ, this Bulletin, vol. 1 (1995), no. 1, pp. 75–84.Google Scholar
[45] Steel, John R., Projectively well-ordered inner models, Annals of Pure and Applied Logic, vol. 74 (1995), no. 1, pp. 77–104.CrossRefGoogle Scholar
[46] Steel, John R., PFA implies ADL(ℝ) , The Journal of Symbolic Logic, vol. 70 (2005), no. 4, pp. 1255–1296.CrossRefGoogle Scholar
[47] Steel, John R., A stationary-tower-free proof of the derived model theorem, Advances in logic, Contemporary Mathematics, vol. 425, American Mathematical Society, Providence, RI, 2007, pp. 1–8.CrossRefGoogle Scholar
[48] Steel, John R., Derived models associated to mice, Computational prospects of infinity. Part I. Tutorials, Lecture Notes Series. Institute for Mathematical Sciences. National University of Singapore, vol. 14, World Science Publications, Hackensack, NJ, 2008, pp. 105–193.CrossRefGoogle Scholar
[49] Steel, John R., The derived model theorem, Logic Colloquium 2006, Lecture Notes in Logic, Association for Symbolic Logic, Chicago, IL, 2009, pp. 280–327.CrossRefGoogle Scholar
[50] Steel, John R., An outline of inner model theory, Handbook of set theory (Foreman, Matthew and Kanamori, Akihiro, editors), vol. 3, Springer, Dordrecht, 2010, pp. 1595–1684.CrossRefGoogle Scholar
[51] Steel, John R.. The triple helix, Workshop on Set Theory and the Philosophy of Mathematics, University of Pennsylvania, 2010, slides available at http://math.ucla.edu/~steel/.Google Scholar
[52] Steel, John R. and Woodin, Hugh, Exploring HOD, The Cabal Seminar (Kechris, Alexander S., Löwe, Benedikt, and Steel, John R., editors), vol. IV, to appear.Google Scholar
[53] Trang, Nam, HOD in natural models of AD+, in preparation.Google Scholar
[54] Viale, Matteo and Weiss, Christoph, On the consistency strength of the proper forcing axiom, Advances in Mathematics, vol. 228 (2011), no. 5, pp. 2672–2687.CrossRefGoogle Scholar
[55] Woodin, W. Hugh, The axiom of determinacy, forcing axioms, and the nonstationary ideal, de Gruyter Series in Logic and its Applications, vol. 1, Walter de Gruyter, Berlin, 1999.CrossRefGoogle Scholar
[56] Woodin, W. Hugh, Strong axioms of infinity and the search for V, Proceedings of the International Congress of Mathematicians, vol. I. Hindustan Book Agency, New Delhi, 2010, pp. 504–528.Google Scholar
[57] AIM workshop on recent advances in core model theory, Open problems, available at http://www.math.cmu.edu/users/eschimme/AIM/problems.pdf.Google Scholar
[58] Zhu, Yizheng, The derived model theorem II, available at http://math.unimuenster.de/logik/Personen/rds/core_model_induction_and_hod_mice.html.Google Scholar
[59] Zhu, Yizheng, Realizing an AD+ model as a derived model of a premouse, in preparation.Google Scholar