# On the strength of second-order set theories beyond ZFC, PSC-CUNY Research Award grant, 2016

J. D. Hamkins, On the strength of second-order set theories beyond ZFC, PSC-CUNY Research Award grant #69573-00 47, funded for 2016-2017.

Abstract. Professor Hamkins proposes to undertake research in the area of logic and foundations known as set theory, focused on the comparative strengths of several of the second-order set theories upon which several prominent recent research efforts have been based. These theories span the range from ZFC through GBC, plus various comprehension, transfinite recursion or class determinacy principles, up to KM and through the hierarchy to KM+ and beyond. Hamkins’s recent result with Gitman characterizing the precise strength of clopen determinacy for proper class games is a good start for the project, but many open questions remain, and Hamkins outlines various strategies that might solve them.

# On sabbatical, CUNY Fellowship Leave, academic year 2014 – 2015

I shall be on sabbatical from CUNY for the 2014 – 2015 academic year, under the CUNY Fellowship Leave program, devoting myself more fully to my research. I am looking forward to a productive year.  For the latter half of my leave, I shall be Visiting Professor of Philosophy at New York University.

# A new large-cardinal never-indestructibility phenomenon, PSC-CUNY Enhanced Research Award, 2014-2015

J. D. Hamkins, A new large-cardinal never-indestructibility phenomenon, PSC-CUNY Enhanced Research Award 45, funded for 2014-2015.

Abstract. Professor Hamkins proposes to undertake research in the area of logic and foundations known as set theory, focused on the interaction of forcing and large cardinals. In a first project, he will investigate a new large cardinal non-indestructibility phenomenon, recently discovered in his joint work with Bagaria, Tsaprounis and Usuba. In a second project, continuing joint work with Cody, Gitik and Schanker, he will investigate new instances of the identity-crises phenomenon between weak compactness and other much stronger large cardinal notions.

# Weak embedding phenomena in $\omega_1$-like models of set theory, Collaborative Incentive Research Grant award, 2013-2014

V. Gitman, J. D. Hamkins and T. Johnstone, “Weak embedding phenomena in $\omega_1$-like models of set theory,” Collaborative Incentive Research Grant award program, CUNY, 2013-2014.

Summary.  We propose to undertake research in the area of mathematical logic and foundations known as set theory, investigating a line of research involving an interaction of ideas and methods from several parts of mathematical logic, including set theory, model theory, models of arithmetic and computability theory. Specifically, the project will be to investigate the recently emerged weak embedding phenomenon of set theory, which occurs when there are embeddings between models of set theory (using the model-theoretic sense of embedding here) in situations where there can be no $\Delta_0$-elementary embedding. The existence of the phenomenon was established recently by Hamkins, who showed that every countable model of set theory, including every countable transitive model, is isomorphic to a submodel of its own constructible universe and thus has such a weak embedding into its constructive universe. In this project, we take the next logical step by investigating the weak embedding phenomena in $\omega_1$-like models of set theory. The study of $\omega_1$-like models of set theory is significant both because these models exhibit interesting second order properties and because their construction out of elementary chains of countable models directs us to create structurally rich countable models.

# Research in set theory, PSC-CUNY Enhanced Research Award, 2011 – 2012

J. D. Hamkins, Research in set theoryPSC-CUNY Enhanced Research Award 42, 2011-2012.

Abstract.  The Principal Investigator, Professor Hamkins, pursues an active research program, with a stream of scholarly publications, international invitations to speak and distinguished grants, and is active in graduate education, currently supervising four PhD students. Professor Hamkins proposes to undertake research in the area of mathematical logic known as set theory, pursuing several projects unified by the classical Boolean ultrapower construction. Specifically, he seeks to investigate the extent to which various large cardinal extender embeddings are realized as Boolean ultrapowers, with their accompanying canonical generic objects, and to investigate the generalized Bukovsky-Dehornoy phenomenon, among other applications of the Boolean ultrapower. Professor Hamkins has a solid publication record on the topics broadly surrounding the proposed research, and has been a leading researcher on the

particular topic proposed.

# Research in set theory, NSF program grant, 2008- 2012

J. D. Hamkins, Research in Set Theory, National Science Foundation, NSF DMS 0800762, June 1, 2008 — May 31, 2012.

Summary abstract: Professor Hamkins will undertake research in the area of mathematical logic known as set theory, pursuing several projects that appear to be ripe for progress. First, the theory of models of arithmetic, usually considered to stand somewhat apart from set theory, has several fundamental questions exhibiting a deep set-theoretic nature, and an inter-speciality approach now seems called for. The most recent advances on Scott’s problem, for example, involve a sophisticated blend of techniques from models of arithmetic and the Proper Forcing Axiom.  Second, large cardinal indestructibility lies at the intersection of forcing and large cardinals, two central concerns of contemporary set-theoretic research and the core area of much of Professor Hamkins’s prior work, and recent advances have uncovered a surprisingly robust new phenomenon for relatively small large cardinals. The strongly unfoldable cardinals especially have served recently as a surprisingly efficacious substitute for supercompact cardinals in various large cardinal phenomena, including indestructibility and the consistency of fragments of the Proper Forcing Axiom. Third, Professor Hamkins will investigate questions in the emerging set-theoretic focus on second and higher order features of the set-theoretic universe.

This research in mathematical logic and set theory concentrates on topics at the foundations of mathematics, exploring the nature of mathematical infinity and the possibility of alternative mathematical universes. Our understanding of mathematical infinity, fascinating mathematicians and philosophers for centuries, has now crystallized in the large cardinal hierarchy, and a central concern of Professor Hamkins’ research will be to investigate how large cardinals are affected by forcing, the technique invented by Paul Cohen by which set theorists construct alternative mathematical universes. The diversity of these universes is astonishing, and set theorists are now able to construct models of set theory to exhibit precise pre-selected features.

In his final project, Professor Hamkins will pursue research aimed at an understanding of the most fundamental relations between the universe and these alternative mathematical worlds.

# CUNY Collaboration in Mathematical Logic, CUNY Collaboration Incentive grant, 2005 – 2007

CUNY Collaboration in Mathematical Logic (6 PIs), CUNY Collaboration Incentive grant, 2005 – 2007.