Supercompactness can be equiconsistent with measurability. (English) Zbl 1529.03256

Trang, Nam

Notre Dame J. Formal Logic 62, No. 4, 593-618 (2021).

Summary: The main result of this paper, built on previous work by the author and T. M. Wilson [“Supercompact measures on \(\wp_{\omega_1}(\wp (\mathbb{R}))\)”, Preprint], is the proof that the theory “\(\mathsf{AD}_{\mathbb{R}}+\mathsf{DC}+\) there is an \(\mathbb{R}\)-complete measure on \(\Theta\)” is equiconsistent with “\( \mathsf{ZF}+\mathsf{DC}+\mathsf{AD}_{\mathbb{R}}+\) there is a supercompact measure on \(\wp_{\omega_1}(\wp (\mathbb{R}))+\Theta\) is regular.” The result and techniques presented here contribute to the general program of descriptive inner model theory and in particular, to the general study of compactness phenomena in the context of \(\mathsf{ZF}+\mathsf{DC}\).

MSC:

03E45	Inner models, including constructibility, ordinal definability, and core models
03E60	Determinacy principles
03E15	Descriptive set theory

Keywords:

descriptive set theory; mouse; hod mouse; inner model theory; supercompactness on \(\omega_1\)

References:

[1]	Atmai, R., and G. Sargsyan, “Hod up to \[{\mathsf{AD}_{\mathbb{R}}}+\theta\] is measurable,” Annals of Pure and Applied Logic, vol. 170 (2019), pp. 95-108. · Zbl 1537.03056 · doi:10.1016/j.apal.2018.08.013
[2]	Becker, H., “\[ \mathsf{AD}\] and the supercompactness of \[{\text{\aleph }_1} \],” Journal of Symbolic Logic, vol. 46 (1981), pp. 822-42. · Zbl 0498.03044 · doi:10.2307/2273231
[3]	Caicedo, A. E., P. Larson, G. Sargsyan, R. Schindler, J. Steel, and M. Zeman, “Square principles in \[{\mathbb{P}_{\max }}\] extensions,” Israel Journal of Mathematics, vol. 217 (2017), pp. 231-61. · Zbl 1368.03048 · doi:10.1007/s11856-017-1444-8
[4]	Ikegami, D., and N. Trang, “On supercompactness of \[{\omega_1}\],” preprint, arXiv:1904.01815v1 [math.LO].
[5]	Jackson, S., “Structural consequences of \[\mathsf{AD} \],” pp. 1753-876 in Handbook of Set Theory, Vols. 1, 2, 3, edited by M. Foreman and A. Kanamori, Springer, Dordrecht, 2010. · Zbl 1198.03056 · doi:10.1007/978-1-4020-5764-9_22
[6]	Koellner, P., and W. H. Woodin, “Large cardinals from determinacy,” pp. 1951-2119 in Handbook of Set Theory, Vols. 1, 2, 3, edited by M. Foreman and A. Kanamori, Springer, Dordrecht, 2010. · Zbl 1198.03072 · doi:10.1007/978-1-4020-5764-9_24
[7]	Neeman, I., and J. Zapletal, “Proper forcing and \[L(\mathbb{R})\],” Journal of Symbolic Logic, vol. 66 (2001), pp. 801-10. · Zbl 0988.03079 · doi:10.2307/2695045
[8]	Sargsyan, G., Hod Mice and the Mouse Set Conjecture, vol. 236 of Memoirs of the American Mathematical Society, American Mathematical Society, Providence, 2014. · Zbl 1403.03110 · doi:10.1090/memo/1111
[9]	Sargsyan, G., “Covering with universally Baire operators,” Advances in Mathematics, vol. 268 (2015), pp. 603-65. · Zbl 1345.03094 · doi:10.1016/j.aim.2014.09.016
[10]	Sargsyan, G., and N. Trang, “The largest Suslin axiom,” preprint, 2018, http://math.unt.edu/ ntrang/lsa.pdf.
[11]	Solovay, R., “The independence of \[\mathsf{DC}\] from \[\mathsf{AD} \],” pp. 171-83 in Cabal Seminar 76-77 (Caltech-UCLA, 1976-77), edited by A. S. Kechris and Y. N. Moschovakis, vol. 689 of Lecture Notes in Mathematics, Springer, Berlin, 1978. · Zbl 0388.03022
[12]	Steel, J. R., “Closure properties of pointclasses,” pp. 147-63 in Cabal Seminar 77-79 (Caltech-UCLA, 1977-79), edited by A. S. Kechris, D. A. Martin, and Y. N. Moschovakis, vol. 839 of Lecture Notes in Mathematics, Springer, Berlin, 1981. · Zbl 0496.03033
[13]	Steel, J. R., “Scales in \[K(\mathbb{R})\],” pp. 176-208 in Games, Scales, and Suslin Cardinals: The Cabal Seminar, Vol. I, edited by A. S. Kechris, B. Löwe, and J. R. Steel, vol. 31 of Lecture Notes in Logic, Association for Symbolic Logic, Chicago, 2008. · Zbl 1167.03032 · doi:10.1017/CBO9780511546488.011
[14]	Steel, J. R., “Remark on a paper by Sargsyan,” handwritten notes, 2013, https://math.berkeley.edu/ steel/papers/remarks.sargsyan.pdf.
[15]	Steel, J. R., and N. Trang, “\[{\mathsf{AD}^+} \], derived models, and \[{\Sigma_1} \]-reflection,” preprint, http://math.unt.edu/ ntrang (accessed 27 October 2021).
[16]	Trang, N., “Determinacy in \[L(\mathbb{R},\upmu )\],” Journal of Mathematical Logic, vol. 14 (2014), art. ID 1450006. · Zbl 1338.03100 · doi:10.1142/S0219061314500068
[17]	Trang, N., “HOD in natural models of \[{\mathsf{AD}^+} \],” Annals of Pure and Applied Logic, vol. 165 (2014), pp. 1533-56. · Zbl 1354.03067 · doi:10.1016/j.apal.2014.04.006
[18]	Trang, N., “Derived models and supercompact measures on \[{\wp_{{\omega_1}}}(\wp (\mathbb{R}))\],” Mathematical Logic Quarterly, vol. 61 (2015), pp. 56-65. · Zbl 1339.03041 · doi:10.1002/malq.201300061
[19]	Trang, N., “Structure theory of \[L(\mathbb{R},\upmu )\] and its applications,” Journal of Symbolic Logic, vol. 80 (2015), pp. 29-55. · Zbl 1373.03107
[20]	Trang, N., and T. Wilson, “Supercompact measures on \[{\wp_{{\omega_1}}}(\wp (\mathbb{R}))\],” preprint, http://math.unt.edu/ ntrang (accessed 27 October 2021).
[21]	Wilson, T., “Contributions to descriptive inner model theory,” PhD dissertation, University of California, Berkeley, 2012.