Trouble in Cantor´s Paradise

In the late 19th century, Cantor laid the grounds for modern set theory with his proof that there are different sizes of infinityin particular, there are more real numbers than whole numbers. His research was initially met with some resistance, but Hilbert defended him, famously declaring, No one shall expel us from the paradise which Cantor has created for us. Since Cantors work, set theorists have been interested in understanding what distinct properties the different infinities enjoy and how they relate to each other. A common theme is to take well-known properties of the smallest infinity (the set of whole numbers) and ask whether higher infinities can have similar features. Sometimes this leads to a large cardinal, a number so large that it cannot be proven to exist in the standard axiom system ZFC. Other times, we find a property that relatively small infinities can have but which they cannot be proven to have, precisely because it carries some trace of a large cardinal. These large-cardinal properties of small infinites have been heavily studied because they are natural notions that have many consequences for ordinary mathematical structures like the real numbers, collections of functions on real numbers, and so on. Some set theorists have suggested adopting axioms asserting that large-cardinal properties hold quite frequently in the mathematical universe. However, recent work has shown that the surrounding terrain is somewhat treacherous; various forms of these properties sometimes come into conflict with each other in unexpected ways. This project aims to further map out the extent of these tensions, as well as threads of harmony yet to be discovered. The focus is on the interactions among three kinds of phenomena around successor cardinals: saturated ideals, Changs Conjecture, and the tree property.

In the late 19th century, Cantor laid the grounds for modern set theory with his proof that there are different sizes of infinity--in particular, there are more real numbers than natural numbers. His research was initially met with some resistance, but Hilbert defended him, famously declaring, "No one shall expel us from the paradise which Cantor has created for us." Since Cantor's work, set theorists have been interested in understanding what distinct properties the different infinities enjoy and how they relate to each other. A common theme is to take well-known properties of the smallest infinity (the set of natural numbers) and ask whether higher infinities can have similar features. Sometimes this leads to a large cardinal, a number so large that it cannot be proven to exist in the standard axiom system ZFC. Other times, we find a property that relatively small infinities can have but which they cannot be proven to have, precisely because it carries some trace of a large cardinal. These largeness properties of small infinites have been heavily studied because they are natural notions that have many consequences for ordinary mathematical structures like the real numbers, collections of functions on real numbers, and so on. Some set theorists have suggested adopting axioms asserting that largeness properties hold quite frequently in the mathematical universe. However, recent work has shown that the surrounding terrain is somewhat treacherous; various forms of these properties sometimes come into conflict with each other in unexpected ways. This project helped to clarify the this picture in two directions. On the one hand, we found further tensions between two different kinds of largeness properties-- compactness and hugeness-- at the second uncountable cardinal. We also found that very strong hugeness axioms cannot coexist at adjacent cardinals. On the other hand, we obtained a harmonious synthesis, showing the consistency of every successor of a regular cardinal being "minimally generically almost-huge", which means that all of these sizes of infinity can enjoy simultaneously a simple and powerful largeness property. This answered some questions asked by Foreman in the early days of the study of these notions and led to the solution of several open problems in infinite combinatorics, with applications to graph colorings and homological algebra.