Combinatorial 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 combinatorial scales (but usually they’re simply called scales as well, however).

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Scales 101 – part III: moving to L(R)

1200px-Brass_scales_with_cupped_trays

The last two posts covered the ‘classical’ theory of scales, meaning the characterisation of the scaled pointclasses in the projective hierarchy. Noting that \bf\Sigma^1_n=\bf\Sigma_n^{J_0(\mathbb R)} and \bf\Pi^1_n=\bf\Pi_n^{J_0(\mathbb R)}, the natural generalisation of this characterisation is then to figure out which of the \bf\Sigma_n^{J_\alpha(\mathbb R)} and \bf\Pi_n^{J_\alpha(\mathbb R)} classes are scaled, for \alpha>0. 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 \textsf{AD}^{L(\mathbb R)}.

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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.

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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.

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