The revolutionary work of Cohen and Gödel on Hilbert’s First Problem led to the development of set-theoretic techniques that made it possible to show that various natural mathematical questions are not answered by the standard axiomatization of mathematics provided by the Zermelo–Fraenkel Axioms of Set Theory together with the Axiom of Choice (ZFC). This development initiated the search for the right axiomatization of mathematics with researchers seeking for intrinsically justified extensions of ZFC that settle important questions left open by these axioms. Large cardinal axioms, postulating the existence of cardinals having structural properties that make them so large that their existence implies the consistency of ZFC, play a central role in this search, because they allow us to measure the consistency strength of extensions of ZFC and order them into a linear hierarchy. In addition, axioms of this form solve many questions left open by ZFC in the desirable way, and therefore they are themselves strong candidates for the correct axiomatization of mathematics.
Despite this central role in modern set theory, large cardinals are still surrounded by many open fundamental questions. In particular, there is no widely accepted definition of what a large cardinal actually is and, lacking such a definition, it seems impossible to actually develop a general theory of large cardinals that allows proofs of their observed properties. Moreover, although large cardinal axioms were shown to have many desirable consequences, the question whether they are true and should be added to the standard axiomatization of mathematics remains widely open.
Besides Cantor’s Continuum Hypothesis, several important questions coming from very different parts of pure mathematics were shown to be independent of the axioms of ZFC during the last seventy years. Prominent examples of such questions are given by Shelah’s solution of the Whitehead Problem in group theory and Farah’s work on the automorphism group of the Calkin algebra in functional analysis. Results of this form demonstrate the importance of the search for new axioms for mathematics and underline the necessity to increase our understanding of large cardinals to evaluate candidates for such axioms.
Moreover, new strong reasons to add large cardinal assumptions to the axioms of mathematics were recently given by solutions to long-standing open questions in other parts of pure mathematics, like homotopy theory and commutative algebra, based in these assumption and, since these results unveiled various fruitful possibilities for further applications, research in these directions has recently flourished.
The aim of this project was to evolve our understanding of large cardinals in three directions. First, in order to work towards the goal of finding a widely accepted definition of large cardinal axioms, we worked on the development of uniform frameworks for large cardinals and their ordering under both direct implications and consistency strength. Second, we studied the set-theoretic consequences of large cardinal axioms, focusing on their interaction with set-theoretic definability. Third, we worked to widen the applications of large cardinals outside of set theory, focusing on applications of Vopenka’s Principles and its variations. As we will outline below, the results obtained in this project allow us to conclude that all three goals were achieved.
