PreprintCC BY 4.0arXiv
The Cone Fixing the Reals of $\mathrm{HOD}^{M_{2n}(x)}
Abstract
For a cone of reals x, the reals of HODL[x] are precisely the reals in Q3, or equivalently, the set of reals in M1, the minimal proper class mouse with a Woodin cardinal. We show that if z is any real coding M1|δ+M1, then z is the base of such a cone (so M1# is such a base). We also establish a kind of optimality for this result, by identifying a real x such that M1⊆ L[x] but ℝ∩HODL[x]not⊆ M1, although there is a base for the cone in a generic extension of L[x]. We prove analogous results for M2n(x) for all n, with the optimality result holding for all n.
Related
BibTeX
@misc{schlutzenberg2026cone,
title = {{The Cone Fixing the Reals of $\mathrmHOD^M_2n(x)}},
author = {Farmer Schlutzenberg and Connor Watson},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06032},
eprint = {2611.06032},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06032},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - The Cone Fixing the Reals of $\mathrm{HOD}^{M_{2n}(x)}
AU - Farmer Schlutzenberg
AU - Connor Watson
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06032
UR - https://arxiv.org/abs/2611.06032
AB - For a cone of reals x, the reals of HOD^L[x] are precisely the reals in Q_3, or equivalently, the set of reals in M_1, the minimal proper class mouse with a Woodin cardinal. We show that if z is any real coding M_1|δ^+M_1, then z is the base of such a cone (so M_1^# is such a base). We also establish a kind of optimality for this result, by identifying a real x such that M_1⊆ L[x] but ℝ∩HOD^L[x]not⊆ M_1, although there is a base for the cone in a generic extension of L[x]. We prove analogous results for M_2n(x) for all n, with the optimality result holding for all n.
ER -