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.
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).
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.
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 .
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.