The strong reflecting property and Harrington's Principle

13Note that the proof of [8, Theorem 18.20], as opposed to the proof of Theorem 3.3 above, is

not done in Z4.

14Examples of large cardinal notions compatible with L: inaccessible cardinal,reflecting car-

nal, ineffable cardinal, 1-iterable cardinal, remarkable cardinal, 2-iterable cardinal and ω-Erdo¨s

cardinal. [1] Yong Cheng, Forcing a set model of Z3 + Harrington's Principle, To appear in Mathematical

Logic Quarterly. [2] Yong Cheng and Ralf Schindler, Harrington's Principle in higher order arithmetic, To appear

in The Journal of Symbolic Logic. [3] James Cummings, Iterated Forcing and Elementary Embeddings, Chapter 12 in Handbook of

Set Theory, Edited by Matthew Foreman and Akihiro Kanamori, Springer, Berlin, 2010. [4] Keith J.Devlin, Constructibility, Springer, Berlin, 1984. [5] Sy D. Friedman, Constructibility and Class Forcing, Chapter 8 in Handbook of Set Theory,

Edited by Matthew Foreman and Akihiro Kanamori, Springer, Berlin, 2010. [6] Victoria Gitman, Joel David Hamkins, Thomas A. Johnstone, What is the theory ZF C

without Powerset? See http://arxiv.org/abs/1110.2430 [7] L.A. Harrington, Analytic determinacy and 0♯, The Journal of Symbolic Logic, 43(1978),

685-693. [8] Thomas J.Jech, Set Theory, Third millennium edition, revised and expanded, Springer,

Berlin, 2003. [9] J. D. Hamkins, G. Kirmayer, and N. L. Perlmutter, Generalizations of the Kunen inconsis-

tency, Annals of Pure and Applied Logic, vol. 163, iss. 12, pp. 1872-1890, December 2012. [10] Akihiro Kanamori, The Higher Infinite: Large Cardinals in Set Theory from Their Begin-

nings, Springer Monographs in Mathematics, Springer, Berlin, 2003, Second edition. [11] Kenneth Kunen, Set Theory: An Introduction to Independence Proofs, North-Holland, Am-

sterdam, 1980. [12] William J. Mitchell, Beginning Inner Model Theory, Chapter 17 in Handbook of Set Theory,