{"paper":{"title":"Combinatorics in Higher Solovay Models","license":"http://creativecommons.org/licenses/by/4.0/","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,,","cross_cats":[],"primary_cat":"math.LO","authors_text":"Alejandro Poveda, Sebastiano Thei","submitted_at":"2025-09-23T13:39:14Z","abstract_excerpt":"We construe the singular-cardinal analogue of the classical Solovay model. 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"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"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. This is the first Solovay-type model at the first singular cardinal.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The consistency of ZFC together with the appropriate large cardinal axioms is assumed in order to produce the model; without this background consistency the forcing or inner-model construction that arranges the five listed properties at aleph_omega cannot be carried out.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"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,,","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"31b96e13ccdc6b383ae41d56bfb1eec36ba8fe97935e930af10c7f4fa2b25213"},"source":{"id":"2509.18991","kind":"arxiv","version":4},"verdict":{"id":"c70062ff-2303-477e-a56f-5fe06edc2c38","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-18T14:31:03.874693Z","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. This is the first Solovay-type model at the first singular cardinal.","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","weakest_assumption":"The consistency of ZFC together with the appropriate large cardinal axioms is assumed in order to produce the model; without this background consistency the forcing or inner-model construction that arranges the five listed properties at aleph_omega cannot be carried out.","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,,"},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2509.18991/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":2,"snapshot_sha256":"d2dbda6d2797067943b895699bf50509fa0b46ac4372fd2b909cd41198e76476"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}