The Alfred Tarski Lectures

Following the death of Group founder Alfred Tarski in 1983, an endowment fund was established in his memory. Using income from this fund, a series of annual Alfred Tarski Lectures was inaugurated in 1989. Each spring an outstanding scholar in a field to which Tarski contributed is selected to come to Berkeley to meet with faculty and students and to deliver several lectures. (List of past Tarski lectures.)

The Thirty-sixth Annual Alfred Tarski Lectures

Grigor Sargsyan (Institute of Mathematics, Polish Academy of Sciences)

Gödel’s Program and Two Dogmas of Set Theory

Gödel’s Program

In set theory, Gödel’s program is a response to the independence phenomenon; its aim is to remove independence from set theory by passing to stronger and stronger theories of infinity that gradually decide all undecidable statements. In this talk, we will present the current state of the program and explain how it naturally leads to studying the canonical structures existing in the universe that form the core part of any foundational theory, such as Martin’s Maximum or the Axiom of Determinacy. It has also led set theorists to particularly rigid beliefs about the complexity of their guiding principles. In subsequent talks, we will examine some of these dogmas and their current status.

The Nairian Models Perspective

Gödel’s program rests on the idea that large cardinal axioms form a hierarchy ordered by consistency strength that covers every level of the consistency strength hierarchy. This large cardinal-centric view of set theory has led to deeply held beliefs about the large cardinal strength of various natural, mathematically useful, and rich theories like those mentioned above: Martin’s Maximum, theories postulating the determinacy of various games, theories postulating the existence of generic elementary embeddings, and others. In this talk, we will present a modern perspective based on new types of models of determinacy called Nairian Models. We will show that several of the aforementioned beliefs or conjectures were misguided and provide alternative views. Specifically, we will consider our first dogma: that the failure of Jensen’s and Todorčević’s square principles at all uncountable cardinals has a very large consistency strength.

Inner Models and the Powerset Operation

The Inner Model Program, which is a key component of Gödel’s Program, aims to build canonical inner models for large cardinal axioms. The least canonical inner model is Gödel’s constructible universe L, and one of the desired features of these inner models is L-likeness. Unlike the cumulative hierarchy, the L-hierarchy evolves by restricting only to definable sets, which leads to a very restricted notion of the powerset operation. This has led to a commonly held belief that the powerset operation inside inner models is simple, and therefore, the universe of sets cannot be an inner model—otherwise, everything would be very simple. In fact, we will show that, surprisingly, the most robust generically absolute notion of definability—universally Baire definability—is not powerful enough to capture the powerset operation in very small inner models. We will then explore generically absolute notions of definability discovered in recent years that go beyond universally Baire sets. The series of talks will conclude with a view of the universe of sets based on Nairian Models. All three talks will be aimed at a general audience.