Mouse scales
Farmer Schlutzenberg
TL;DR
The paper develops a framework to derive scales and prewellorderings directly from mouse existence hypotheses, avoiding determinacy. It generalizes the Martin–Solovay approach by using inner-model theory, iterative background constructions, and comparisons among background premice to produce scales for a wide range of pointclasses, including $\Pi^1_3$ and well beyond the projective hierarchy. Central innovations include the $L[\mathbb{E},x]$-style background construction, the $\mathbb{D}$-construction and its role in semiscale-to-scale conversion, and detailed iterability/condensation analyses for $1$-small and $Q$-mice. The work yields both classical scale results without determinacy and broad new pointclasses with the scale property, unlocking further uniformization and structural results under large-cardinal hypotheses and mouse existence. The methods blend inner model theory, fine structure, and novel comparison techniques to achieve optimal definability and robustness of scales across an expansive landscape of pointclasses.
Abstract
We give a construction of scales (in the descriptive set theoretic sense) directly from mouse existence hypotheses, without using any determinacy arguments. The construction is related to the Martin-Solovay construction for scales on $Π^1_2$ sets. The prewellorders of the scales compare reals $x$ and $y$ by comparing features of certain kinds of fully backgrounded $L[E,x]$- and $L[E,y]$-constructions executed in mice $P$ with $x,y \in P$. In this way we produce an inner model theoretic proof of the scale property for many pointclasses, for which the scale property was classically established using determinacy arguments (for example, $Π^1_3$). Moreover, it also yields many further pointclasses with the scale property, for example intermediate between $Π^1_{2n+1}$ and $Σ^1_{2n+2}$, and also instances of complexity well beyond projective.