When working with most of modern set theory we tend to transcend ZFC, always working with some strong background hypothesis, whether it being the existence of some elementary embedding, a colouring for some partition property, a generic for some uncountable poset or something completely different. When it comes to *using* these strong hypotheses in mainstream mathematics it seems that we hit a brick wall, as most of our strong hypotheses don’t easily translate to the language of everyday mathematics.

# Author: Dan Saattrup Nielsen

# Pcf scales and squares

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 *pfc 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.