Talk: Jónsson cardinals – a deceptive large cardinal axiom


I gave a contributed talk at the British Logic Colloquium at the University of Sussex on September 7.

Notes and slides can be found on my talks page. Here’s an abstract:

Does every structure have a proper elementary substructure of the same size? This question — whose answer is false — led to the notion of a Jónsson cardinal, being the cardinals for which this is true. These cardinals turn out to exhibit strange behaviour compared to other large cardinals, as the theory “ZFC + there is a Jónsson cardinal” is quite strong, but a Jónsson cardinal itself is ostensibly very weak in terms of direct implication. I will talk a bit about this asymmetry in strength and zoom in on the problem of determining the direct implication strength, leading to a special relationship between Jónsson cardinals and weakly compact cardinals.

This is partially based on the content of two of my previous blog posts about Jónsson successors and approximations to weak compactness.