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.