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 sets. The prewellorders of the scales compare reals and by comparing features of certain kinds of fully backgrounded - and -constructions executed in mice with . 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, ). Moreover, it also yields many further pointclasses with the scale property, for example intermediate between and , and also instances of complexity well beyond projective.
Cite
@article{arxiv.2310.19764,
title = {Mouse scales},
author = {Farmer Schlutzenberg},
journal= {arXiv preprint arXiv:2310.19764},
year = {2025}
}
Comments
Preliminary draft v3; some proofs incomplete. 139 pages. Corrected acknowledgements and updated references