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Starting with large cardinal assumptions in the realm of supercompactness, we show that the our inner model captures a substantial portion of the combinatorics of $L(\\mathcal{P}(\\kappa))$ that are typically implied by Woodin's axiom $I_0$. Among other things, we show that in our higher Solovay model there are no $\\kappa^+$-sequences of distinct members of $\\mathcal{P}(\\kappa)$ and that Shelah's approachability property $\\AP_\\kappa$ fails. We prove that every set in our inner model satisfies a singular analogue of the co","authors_text":"Alejandro Poveda, Sebastiano Thei","cross_cats":[],"headline":"Assuming the consistency of ZFC with large cardinals, a model exists in which aleph_omega is a strong limit and L(P(aleph_omega)) satisfies the aleph_omega-perfect set property for all subsets of sequences, has no scale, fails SCH and AP,,","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.LO","submitted_at":"2025-09-23T13:39:14Z","title":"Combinatorics in Higher Solovay Models"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2509.18991","kind":"arxiv","version":4},"verdict":{"created_at":"2026-05-18T14:31:03.874693Z","id":"c70062ff-2303-477e-a56f-5fe06edc2c38","model_set":{"reader":"grok-4.3"},"one_line_summary":"From large cardinals, constructs a model where aleph_omega is strong limit, L(P(aleph_omega)) has aleph_omega-PSP, no scales, SCH and AP fail, TP holds at aleph_omega+1, answering Woodin's question on SCH vs AP.","pipeline_version":"pith-pipeline@v0.9.0","pith_extraction_headline":"Assuming the consistency of ZFC with large cardinals, a model exists in which aleph_omega is a strong limit and L(P(aleph_omega)) satisfies the aleph_omega-perfect set property for all subsets of sequences, has no scale, fails SCH and AP,,","strongest_claim":"Assuming the consistency of ZFC with appropriate large cardinal axioms, there is a model of ZFC in which aleph_omega is a strong limit cardinal and L(P(aleph_omega)) satisfies: every A subset (aleph_omega)^omega has the aleph_omega-PSP, there is no scale at aleph_omega, SCH fails at aleph_omega, AP fails at aleph_omega, and TP holds at aleph_omega+1. 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