# Generalised square inaccessibility

Jensen’s square principle $\Box_\kappa$ has proven very useful in measuring the non-compactness of various successor cardinals as well as being an essential tool in finding new lower bounds for forcing axioms like the Proper Forcing Axiom. It should be noted however, that $\Box_\kappa$ is not really about $\kappa$, but about $\kappa^+$. To remedy this confusion, Caicedo et al (’17) came up with the term square inaccessible instead, where $\kappa^+$ is square inaccessible if $\Box_\kappa$ fails. It seems as though we can only talk about successor cardinals being square inaccessible then, but results from Krueger (’13) and Todorčević (’87) allow us generalise this to all uncountable regular cardinals. I’ll introduce this generalisation here and note that the celebrated result of Jensen (’72), stating that there aren’t any successor square inaccessible cardinals in L, does not hold for all cardinals.

# Partition relations, Suslin trees and Jónsson cardinals — an interplay of open problems

I’ve previously mentioned the open problem of whether inaccessible Jónssons provably are weakly compact. Here I want to present a few other seemingly completely different open problems in set theory raised by various people and show the interaction between them.

# Descriptive inner model theory – an overview

Inner model theory is all about constructing canonical inner models of set theory that inherits the complexity — i.e. the large cardinals — of the universe. In “classical”, or “pure”, inner model theory it’s clear that there’s been a lot of partial progress towards this goal, as the programme has resulted in explicit constructions of inner models inheriting a lot of the large cardinals present in V. But with the emergence of descriptive inner model theory this is suddenly not as clear. Where are the inner models containing large cardinals? I’ll do my best to give an overview of how this is accomplished and also how large cardinal theories, determinacy theories and arbitrary theories of interest (like forcing axioms) interact with each other.

# 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.

# Three paths to weak compactness

Weakly compact cardinals have a great variety of properties, which are all interesting enough to warrant independent study. These properties include threadabilityreflection and Mahlo properties. Studying these features in isolation leads to interesting (non-)interactions and gives us three distinct hierarchies of large cardinal notions below weakly compacts in terms of direct implication, where in terms of consistency strength two of the hierarchies simply collapse.

# Consistency strength of forcing axioms

Previously I’ve only been talking about large cardinals and determinacy theories as if they were the only consistency hierarchies around. There is another important class of axioms, which has the added benefit of being, dare I say it, more useful to mathematicians not working in set theory. The reason for this is probably that these forcing axioms have a handful of consequences of a non-set theoretic nature, making them easier to apply in (mathematical) practice. When it comes to the consistency strength of these axioms though, things get a lot more hazy: we know very little about the strength of (almost all of) these axioms. I’ll introduce these axioms here and state what is known to date.

# Jónsson successors of singulars

We currently don’t know whether or not there can exist a singular cardinal $\rho$ such that $\rho^+$ is a Jónsson cardinal. I’ll try to survey some of the properties these strange things satisfy and how much is known about the consistency strength of the existence of them.