Conference: Set Theory: Large Cardinals, Forcing and Higher Order Languages

This award a conference taking place at University of California, Irvine from February 9 to February 12, 2026. The title is: Set Theory: Larger Cardinals, Forcing and Higher Order Languages. The meeting is devoted to advances in finding a critical understanding the underlying assumptions and logic that mathematicians use. What separates mathematics from other fields of science is that mathematical facts are known for certain, and not subject to data errors, or a changing environment. The method used for this is that of proofs. For a mathematical fact (such as the Central Limit Theorem) to be known, there has to be a proof of it. But Proofs are divided into assumptions A and consequences B and a proof that A implies B. What constitutes a proof is well understood, but there are many unknown questions about what the appropriate assumptions are. Most of mathematics follows from the collection of assumptions referred to as ZFC. But important questions cannot be settled by ZFC. This meeting studies the relationships between the assumptions that can be added to augment the ZFC assumptions and clarifies what they imply.

There are three well-accepted methods for adding and comparing axiom systems for mathematics. They are: adding Large Cardinal assumptions, using the method of Forcing to alter examples where ZFC plus the additional axioms hold and a method called Inner Model Theory. Some of the assumptions combine both Large Cardinals and Forcing (these are called Strong Forcing Axioms). Much of the conference will be devoted to understanding the relationship between Forcing, Large Cardinals and how the power set operation behaves. This will be related to the use of powerful languages (Strong Logics) to construct inner models, and the construction of mathematical trees with strong properties. Organizers expect to be able to support about ten graduate students and postdocs and similar number of senior mathematicians.

This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

Award Number: 2544999
Principal Investigator: Matthew Foreman
Funds Obligated: $34,826
State: CA