The last few posts I’ve been covering a characterisation of pointclasses that admit scales. To make scale theory even more confusing there’s a completely different notion of scale, which really has nothing to do with our previous one — this one being of a more combinatorial nature. To avoid unnecessary confusion I’ll call these new objects *combinatorial scales* (but usually they’re simply called scales as well, however).

# Talk: Mapping the Ramsey-like cardinals

On December 18, I will be giving an invited talk at the Bonn Logic Seminar.

# Scales 101 – part IV: leaving a gap

So far we’ve characterised the scaled pointclasses among the projective hierarchy as well as establishing Steel’s result that is scaled for *all* such that . We now move on to boldface territory, finishing off this series on scales.

# Scales 101 – part III: moving to L(R)

The last two posts covered the ‘classical’ theory of scales, meaning the characterisation of the scaled pointclasses in the projective hierarchy. Noting that and , the natural generalisation of this characterisation is then to figure out which of the and classes are scaled, for . This is exactly what Steel (’83) did, and I’ll sketch the results leading up to this characterisation in a couple of blog posts. This characterisation is also precisely what’s used in organising the induction in core model inductions up to .

# Scales 101 – part II: where & how?

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.

# Scales 101 – part I: what & why?

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.

# Generalised square inaccessibility

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.