The Strength of Very Large Cardinals

Large Cardinal Hypotheses are currently the main area of investigation for the analysis of the relative consistency of mathematical propositions and possible new axioms for mathematics. We call Very Large Cardinals the few hypotheses that imply the consistency of all the other known Large Cardinals, therefore carrying the most power among all the Large Cardinal and potentially being the most fruitful. Unfortunately there are currently no tools that permit to practically use this strength to achieve results in Set Theory and Mathematics at large. The aim of the project is to fill this gap. The first phase will be dedicated to a technical analysis of the consistency of Very Large Cardinals, mapping the possible structures of a universe containing them and delineating the forcing notions that behave well with them. The second phase will use the achievements from the previous phase to prove results about a generalized Descriptive Set Theory, Topology and Algebra. In the third phase we will try to push even more the Very Large Cardinals, with an analysis of hypotheses even stronger. The project will be run at the Kurt Gödel Research Center.

Large Cardinal Hypotheses are currently the main area of investigation for the analysis of the relative consistency of mathematical propositions and possible new axioms for mathematics. We call Very Large Cardinals the few hypotheses that imply the consistency of all the other known Large Cardinals, therefore carrying the most power among all the Large Cardinal and potentially being the most fruitful. Unfortunately there are currently no tools that permit to practically use this strength to achieve results in Set Theory and Mathematics at large. The aim of the project was to fill this gap. The first phase was dedicated to a technical analysis of the consistency of Very Large Cardinals, mapping the possible structures of a universe containing them and delineating the forcing notions that behave well with them. The second phase consisted in shrinking Very Large Cardinals to accessible small cardinals, reaching results that could solve long-standing open problems and paint a striking picture of sets without the axiom of choice. The project has been run at the Kurt Gödel Research Center.