Identity crises phenomena in the large cardinal hierarchy.

It is well-known that certain mathematical questions cannot be answered on the grounds of the standard foundation of mathematics. Large cardinal axioms constitute a series of postulates about the higher infinite which permit to classify these undecidable problems in a coherent hierarchy way. Specifically, large cardinals together with ZFC (the standard axiomatic of Mathematics) provide a complete classification of all mathematical theories according to the so-called consistency strength. One of the main tenets of modern set theory has been to investigate how the large-cardinal hierarchy is organized across the mathematical universe. To a large extent this hierarchy is nicely disposed and such a disposition is unambiguous (i.e., immune to the independence phenomenon).

In an unexpected turn of events, in the late 70’s Magidor discovered the identity crisis phenomena of the large cardinal hierarchy. Magidor proved that certain strata of the hierarchy are susceptible to be modified via Cohen’s method of forcing. Specifically, he showed that the first strongly compact cardinal can be either the first measurable cardinal or the first supercompact cardinal. It turns out that the first measurable is always much smaller than the first supercompact. These discrepancies on the identity of the first strongly compact cardinal were termed by Magidor the Identity Crisis Phenomenon.

In this talk I plan to provide an introduction to the world of large cardinals keeping an eye on the identity crises phenomena. Time permitting, I’ll present a few recent results answering questions by Magidor and discuss their connection with Woodin’s Ultimate-L Conjecture.

In-person and on Zoom: