Abstract
We investigate the possibilities of global versions of Chang’s Conjecture that involve singular cardinals. We show some mathrm{ZFC} limitations on such principles and prove relative to large cardinals that Chang’s Conjecture can consistently hold between all pairs of limit cardinals below aleph _{omega ^omega }.
Highlights
The Löwenheim–Skolem theorem asserts that for every pair of infinite cardinals κ > μ and every structure A on κ in a countable language, there is a substructure B ⊆ A of size μ
The full Global Chang’s Conjecture is inconsistent, as we show in Theorem 2.8
We investigate other forms of Global Chang’s Conjecture: Definition 1.4 (Singular Global Chang’s Conjecture) We say that the Singular Global
Summary
The Löwenheim–Skolem theorem asserts that for every pair of infinite cardinals κ > μ and every structure A on κ in a countable language, there is a substructure B ⊆ A of size μ. (The consistency of contrary cases is unknown.) This inspires the following bold conjecture: Definition 1.3 (Global Chang’s Conjecture) We say that the Global Chang’s Conjecture holds if for all infinite cardinals μ < κ with cf (μ) cf (κ), (κ+, κ) (μ+, μ). In the paper [6], we showed, assuming the consistency of a huge cardinal, that there is a model of ZFC + GCH in which (κ+, κ) (μ+, μ) holds whenever κ is regular and μ < κ is infinite. We present here a partial result, showing that there is a model in which the Singular Global Chang’s Conjecture holds for cardinals below אωω.
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