Abstract. The principle of covering reflection holds of a cardinal κ if for every structure B in a countable first-order language there is a structure A of size less than κ, such that B is covered by elementary images of A in B. Is there any such cardinal? Is the principle consistent? Does it have large cardinal strength? This is joint work with myself, Nai-Chung Hou, Andreas Lietz, and Farmer Schlutzenberg.
This was a talk for the 2022 Set Theory Conference at Oberwolfach, which was a hybrid of in-person talks and online talks on account of the Covid pandemic. I gave my talk online 10 January 2022.
Abstract: Set theory exhibits a truly robust mutual interpretability phenomenon: in any model of one set theory we can define models of diverse other set theories and vice versa. In any model of ZFC, we can define models of ZFC + GCH and also of ZFC + ¬CH and so on in hundreds of cases. And yet, it turns out, in no instance do these mutual interpretations rise to the level of bi-interpretation. Ali Enayat proved that distinct theories extending ZF are never bi-interpretable, and models of ZF are bi-interpretable only when they are isomorphic. So there is no nontrivial bi-interpretation phenomenon in set theory at the level of ZF or above. Nevertheless, for natural weaker set theories, we prove, including ZFC- without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of ZFC- that are bi-interpretable, but not isomorphic—even $\langle H_{\omega_1},\in\rangle$ and $\langle H_{\omega_2},\in\rangle$ can be bi-interpretable—and there are distinct bi-interpretable theories extending ZFC-. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails. This is joint work with Alfredo Roque Freire.
Note: the conference has been cancelled due to concerns over the Coronavirus-19. (Meanwhile, I have given the talk for the Oxford Set Theory Seminar — see below.)
Abstract: Set theory exhibits a truly robust mutual interpretability phenomenon: in any model of one set theory we can define models of diverse other set theories and vice versa. In any model of ZFC, we can define models of ZFC + GCH and also of ZFC + ¬CH and so on in hundreds of cases. And yet, it turns out, in no instance do these mutual interpretations rise to the level of bi-interpretation. Ali Enayat proved that distinct theories extending ZF are never bi-interpretable, and models of ZF are bi-interpretable only when they are isomorphic. So there is no nontrivial bi-interpretation phenomenon in set theory at the level of ZF or above. Nevertheless, for natural weaker set theories, we prove, including ZFC- without power set and Zermelo set theory Z, there are nontrivial instances of bi-interpretation. Specifically, there are well-founded models of ZFC- that are bi-interpretable, but not isomorphic—even $\langle H_{\omega_1},\in\rangle$ and $\langle H_{\omega_2},\in\rangle$ can be bi-interpretable—and there are distinct bi-interpretable theories extending ZFC-. Similarly, using a construction of Mathias, we prove that every model of ZF is bi-interpretable with a model of Zermelo set theory in which the replacement axiom fails. This is joint work with Alfredo Roque Freire.
