Perfect set dichotomy theorem in generalized Solovay model

2025/12/03 by Hiroshi Sakai, Sakai, Hiroshi, Toshimasa Tanno +1
Mathematics · #Advanced Topology and Set Theory #Mathematical and Theoretical Analysis #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2512.03386

Abstract

We prove that the perfect set dichotomy theorem holds in the Solovay model V((ω^ω)^V[G]). Namely, for every equivalence relation E on ℝ, either ℝ/E is well-orderable or there exists a perfect set consisting of E-inequivalent reals. Furthermore we consider a generalization of the Solovay model for an uncountable regular cardinal μ and show that the perfect set dichotomy theorem for μ^μ also holds in that model. We establish the three element basis theorem for uncountable linear orders in the Solovay model for a weakly compact cardinal, in a general form covering the uncountable case.

Citations

• Comparing forcing approaches to dense ideals
• Cardinality of Wellordered Disjoint Unions of Quotients of Smooth Equivalence Relations
• Perfect subsets of generalized Baire spaces and long games
• Definability and almost disjoint families

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• Combinatorics in Higher Solovay Models
• Higher Solovay Models
• The Baire and perfect set properties at singulars cardinals
• The inclusion relations of the countable models of set theory are all isomorphic
• Independence relations in the Solovay model I