## Notre Dame Journal of Formal Logic

### Large Cardinals, Inner Models, and Determinacy: An Introductory Overview

P. D. Welch

#### Abstract

The interaction between large cardinals, determinacy of two-person perfect information games, and inner model theory has been a singularly powerful driving force in modern set theory during the last three decades. For the outsider the intellectual excitement is often tempered by the somewhat daunting technicalities, and the seeming length of study needed to understand the flow of ideas. The purpose of this article is to try and give a short, albeit rather rough, guide to the broad lines of development.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 56, Number 1 (2015), 213-242.

Dates
First available in Project Euclid: 24 March 2015

https://projecteuclid.org/euclid.ndjfl/1427202981

Digital Object Identifier
doi:10.1215/00294527-2835083

Mathematical Reviews number (MathSciNet)
MR3326596

Zentralblatt MATH identifier
1371.03072

#### Citation

Welch, P. D. Large Cardinals, Inner Models, and Determinacy: An Introductory Overview. Notre Dame J. Formal Logic 56 (2015), no. 1, 213--242. doi:10.1215/00294527-2835083. https://projecteuclid.org/euclid.ndjfl/1427202981

#### References

• [1] Banach, S., “Uber additive massfunktionen in alstrakten Mengen,” Fundamenta Mathematicae, vol. 15 (1930), pp. 97–101.
• [2] Davis, M., “Infinite games of perfect information,” pp. 85–101 in Advances in Game Theory, vol. 52 of Annals of Mathematics Studies, Princeton University Press, Princeton, 1964.
• [3] Devlin, K. J., Constructibility, vol. 6 of Perspectives in Mathematical Logic, Springer, Berlin, 1984.
• [4] Devlin, K. J., and R. B. Jensen, “Marginalia to a theorem of Silver,” pp. 115–42 in $\models$ ISILC Logic Conference (Kiel, 1974), edited by A. Oberschelp, G. H. Müller, and K. Potthoff, vol. 499 of Lecture Notes in Mathematics, Springer, Berlin, 1975.
• [5] Dodd, A. J., and R. B. Jensen, “The core model,” Annals of Mathematical Logic, vol. 20 (1981), pp. 43–75.
• [6] Friedman, H. M., “Higher set theory and mathematical practice,” Annals of Mathematical Logic, vol. 2 (1970), pp. 325–27.
• [7] Gödel, K., The Consistency of the Continuum Hypothesis, vol. 3 of Annals of Mathematics Studies, Princeton Univ. Press, Princeton, 1940.
• [8] Harrington, L., “Analytic determinacy and $0^{\#}$,” Journal of Symbolic Logic, vol. 43 (1978), pp. 685–93.
• [9] Hanf, W., “Incompactness in languages with infinitely long expressions,” Fundamenta Mathematicae, vol. 53 (1964), 325–334.
• [10] Jech, T., Set Theory, 3rd millennium edition, revised and expanded, Springer Monographs in Mathematics, Springer, Berlin, 2003.
• [11] Jensen, R. B., “Inner models and large cardinals,” Bulletin of Symbolic Logic, vol. 1 (1995), pp. 393–407.
• [12] Kanamori, A., The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings, 2nd edition, Springer Monographs in Mathematics, Springer, New York, 2003.
• [13] Kechris, A. S., “Homogenous trees and projective scales,” pp. 33–73 in Cabal Seminar 77–79, edited by D. A. Martin, A. Kechris, and Y. Moschovakis, vol. 839 of Lecture Notes in Mathematics, Springer, Berlin, 1981.
• [14] Koellner, P., “Large cardinals and determinacy,” in Stanford Encyclopedia of Philosophy, 2013, http://plato.stanford.edu/entries/large-cardinals-determinacy.
• [15] Koellner, P., and W. H. Woodin, “Large cardinals from determinacy,” pp. 1951–2120 in Handbook of Set Theory, Vols. 1–3, edited by M. Foreman, A. Kanamori, and M. Magidor, Springer, Dordrecht, 2010.
• [16] Kunen, K., “Some applications of iterated ultrapowers in set theory,” Annals of Mathematical Logic, vol. 1 (1970), pp. 179–227.
• [17] Kunen, K., “Elementary embeddings and infinitary combinatorics,” Journal of Symbolic Logic, vol. 36 (1971), pp. 407–13.
• [18] Magidor, M., “Representing simple sets of ordinals as countable unions of sets in the core model,” Transactions of the American Mathematical Society, vol. 317 (1990), pp. 91–126.
• [19] Martin, D. A., “Measurable cardinals and analytic games,” Fundamenta Mathematicae, vol. 66 (1969), pp. 287–91.
• [20] Martin, D. A., “Borel determinacy,” Annals of Mathematics (2), vol. 102 (1975), 363–71.
• [21] Martin, D. A., “Infinite games,” pp. 363–71 in Lehto and Olli, editors, Proceedings of the International Congress of Mathematicians, vol. 1, Academia Scientiarum Fennica, 1980.
• [22] Martin, D. A., and R. M. Solovay, “A basis theorem for $\Sigma^{1}_{3}$ sets of reals,” Annals of Mathematics (2), vol. 89 (1969), pp 138–59.
• [23] Martin, D. A., and J. R. Steel, “A proof of projective determinacy,” Journal of the American Mathematical Society, vol. 2 (1989), pp. 71–125.
• [24] Mitchell, W. J., “Ramsey cardinals and constructibility,” Journal of Symbolic Logic, vol. 44 (1979), pp. 260–66.
• [25] Mitchell, W. J., “Jónsson cardinals, Erdős cardinals, and the core model,” Journal of Symbolic Logic, vol. 64 (1999), pp. 1065–86.
• [26] Moschovakis, Y. N., Descriptive Set Theory, vol. 100 of Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam, 1980.
• [27] Mycielski, J., and S. Swierczkowski, “On the Lebesgue measurability and the axiom of determinateness,” Fundamenta Mathematicae, vol. 54 (1964), 67–71.
• [28] Sami, R. L., “Analytic determinacy and $0^{\#}$: A forcing-free proof of Harrington’s theorem,” Fundamenta Mathematicae, vol. 160 (1999), pp. 153–59.
• [29] Schimmerling, E., and M. Zeman, “Square in core models,” Bulletin of Symbolic Logic, vol. 7 (2001), pp. 305–14.
• [30] Schimmerling, E., and M. Zeman, “Characterization of $\square _{\kappa}$ in core models,” Journal of Mathematical Logic, vol. 4 (2004), pp. 1–72.
• [31] Scott, D., “Measurable cardinals and constructible sets,” Bulletin de l’Académie Polonaise des Sciences, Série des Sciences Mathématiques, Astrononomiques et Physiques, vol. 9 (1961), pp. 521–24.
• [32] Simpson, S. G., Subsystems of Second Order Arithmetic, Perspectives in Mathematical Logic, Springer, Berlin, 1999.
• [33] Solovay, R., “Measurable cardinals and the continuum hypothesis,” Notices of the American Mathematical Society, vol. 12 (1965), 132.
• [34] Solovay, R. M., W. N. Reinhardt, and A. Kanamori, “Strong axioms of infinity and elementary embeddings,” Annals of Mathematical Logic, vol. 13 (1978), pp. 73–116.
• [35] Steel, J. R., The Core Model Iterability Problem, vol. 8 of Lecture Notes in Mathematical Logic, Springer, Berlin, 1996.
• [36] Ulam, S., “Zur Masstheorie in der allgemeinen Mengenlehre,” Fundamenta Mathematicae, vol. 16 (1930), 140–150.
• [37] Welch, P. D., “Some remarks on the maximality of inner models,” pp. 516–40 in Logic Colloquium ’98 (Prague, 1998), vol. 13 of Lecture Notes in Logic, Association for Symbolic Logic, Urbana, Ill., 2000.
• [38] Woodin, W. H., “The continuum hypothesis, I,” Notices of the American Mathematical Society, vol. 48 (2001), pp. 567–76.
• [39] Woodin, W. H., “The continuum hypothesis, II,” Notices of the American Mathematical Society, vol. 48 (2001), pp. 681–90.