Last time we got an idea of what scales are and why they’re useful. The next questions we then might ask is where do we find them, and how do we create new ones from existing ones? We’ll cover the ‘classical’ answers to these questions here, meaning the ones concerned with the projective hiearchy.
In classical descriptive set theory a need arose to analyse the analytical and projective hierarchies more abstractly, isolating the key structural properties of the various levels. I’ll describe these here and how these concepts can be generalised to the so-called scale property. This is going to be the first post in a series on scales, where we here will focus on what scales are and why they’re useful. The next ones will be concerned with where we find scales and how we construct new scales from existing ones.
Jensen’s square principle 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 is not really about , but about . To remedy this confusion, Caicedo et al (’17) came up with the term square inaccessible instead, where is square inaccessible if 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.
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.
Weakly compact cardinals have a great variety of properties, which are all interesting enough to warrant independent study. These properties include threadability, reflection 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.