Title: Universes of sets

Abstract: As is well-known, all mathematical objects can be coded as sets and thereby all of mathematics can be formally founded in set theory. What is perhaps less well-known is that there are many different models of set theory, each of which is powerful enough to function as a universe of sets and found (most) of mathematics, but these models can have very different properties.

This talk will aim to explore the question: what is a model of set theory? We will learn about Skolem’s paradox, that there are countable models of set theory, even though these countable models think they contain uncountable sets like the set of reals. We will be introduced to transitive models, usually considered to be the best behaved, but also meet ill-founded models, such as models which think ZFC is inconsistent. To conclude we will briefly discuss two positions in the philosophy of set theory: universism, the view that there is a unique maximal universe of sets, and multiversism, the view that there are many equally valid universes of sets.

This is an introductory talk, aimed to be understandable by those with little background in set theory. It is a prequel to my next talk, which is in turn a prequel to my talk after that.

Title: A conceptual overview of forcing

Abstract: Paul Cohen—who visited UH Mānoa in the 1990s—introduced the method of forcing to prove that the failure of the continuum hypothesis is consistent with ZFC, the standard base axioms for set theory. Since then it has become a cardinal tool within set theory, being the main method for proving independence results and even enjoys use in proving ZFC results. In this talk I will give an introduction to forcing, focusing on the big picture ideas.

This talk is a sequel to my previous talk and a prequel to my next talk.

Title: Forcing as a computational process

Abstract: In this talk we will consider computable structure theoretical aspects of forcing. Given an oracle for a countable model of set theory $M$, to what extent can we compute information about forcing extensions $M[G]$? The main theorem I will present gives a robustly affirmative answer in several senses.

* Given an oracle for the atomic diagram of a countable model of set theory $M$, then for any forcing notion $\mathbb P \in M$ we can compute an $M$-generic filter $G \subseteq \mathbb P$.

* From the $\Delta_0$ diagram for $M$ we can moreover compute the atomic diagram of the forcing extension $M[G]$, and indeed its $\Delta_0$ diagram.

* From the elementary for $M$ we can compute the elementary diagram of the forcing extension $M[G]$, and this goes level by level for the $\Sigma_n$ diagrams.

On the other hand, there is no functorial process for computing forcing extensions.

* If ZFC is consistent then there is no computable procedure (nor even a Borel procedure) which takes as input the elementary diagram for a countable model $M$ of ZFC and a partial order $\mathbb P \in M$ and returns a generic $G$ so that isomorphic copies of the same input model always result in the same corresponding isomorphic copy of $G$.

This talk is a sequel to my previous talk. The work in this talk is joint with Joel David Hamkins and Russell Miller.