Place: Fields Institute (Room 210)

Date: May 13, 2016 (13:30-15:00)

Speaker: Peter Koellner

Title: Large Cardinals Beyond Choice

Abstract:

The hierarchy of large cardinals provides us with a canonical means to

climb the hierarchy of consistency strength. There have been many

purported inconsistency proofs of various large cardinal axioms. For

example, there have been many proofs purporting to show that

measurable cardinals are inconsistent. But to date the only proofs

that have stood the test of time are those which are rather

transparent and simple, the most notable example being Kunen’s proof

showing that Reinhardt cardinals are inconsistent. The Kunen result,

however, makes use of AC, and long standing open problem is whether

Reinhardt cardinals are consistent in the context of ZF.

In this talk I will survey the simple inconsistency proofs and then

raise the question of whether perhaps the large cardinal hierarchy

outstrips AC, passing through Reinhardt cardinals and reaching far

beyond. There are two main motivations for this investigation. First,

it is of interest in its own right to determine whether the hierarchy

of consistency strength outstrips AC. Perhaps there is an entire

“choiceless” large cardinal hierarchy, one which reaches new

consistency strengths and has fruitful applications. Second, since the

task of proving an inconsistency result becomes easier as one

strengthens the hypothesis, in the search for a deep inconsistency it

is reasonable to start with outlandishly strong large cardinal

assumptions and then work ones way down. This will lead to the

formulation of large cardinal axioms (in the context of ZF) that start

at the level of a Reinhardt cardinal and pass upward through Berkeley

cardinals (due to Woodin) and far beyond. Bagaria, Woodin, and I

have been charting out this new hierarchy. I will discuss what we have

found so far.