Event Detail

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.