Mouse scales

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 $\Pi^1_2$ 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, $\Pi^1_3$). Moreover, it also yields many further pointclasses with the scale property, for example intermediate between $\Pi^1_{2n+1}$ and $\Sigma^1_{2n+2}$, and also instances of complexity well beyond projective.