TY - JOUR
T1 - Borel sets with large squares by
AU - Shelah, Saharon
PY - 1999
Y1 - 1999
N2 - For a cardinal μ we give a sufficient condition ⊕μ (involving ranks measuring existence of independent sets) for: ⊗μ if a Borel set B ⊆ ℝ × ℝ contains a μ-square (i.e. a set of the form A × A with |A| = μ) then it contains a 2א0-square and even a perfect square, and also for ⊗′μ if ψ ∈ Lω1,ω has a model of cardinality μ then it has a model of cardinality continuum generated in a "nice", "absolute" way. Assuming MA+2א0 > μ for transparency, those three conditions (⊕μ, ⊗μ and ⊗′μ) are equivalent, and from this we deduce that e.g. ∧α<ω1 [2א0 ≥ אα ⇒ ¬⊗אα], and also that min{μ : ⊗μ}, if < 2א0, has cofinality א1. We also deal with Borel rectangles and related model-theoretic problem.
AB - For a cardinal μ we give a sufficient condition ⊕μ (involving ranks measuring existence of independent sets) for: ⊗μ if a Borel set B ⊆ ℝ × ℝ contains a μ-square (i.e. a set of the form A × A with |A| = μ) then it contains a 2א0-square and even a perfect square, and also for ⊗′μ if ψ ∈ Lω1,ω has a model of cardinality μ then it has a model of cardinality continuum generated in a "nice", "absolute" way. Assuming MA+2א0 > μ for transparency, those three conditions (⊕μ, ⊗μ and ⊗′μ) are equivalent, and from this we deduce that e.g. ∧α<ω1 [2א0 ≥ אα ⇒ ¬⊗אα], and also that min{μ : ⊗μ}, if < 2א0, has cofinality א1. We also deal with Borel rectangles and related model-theoretic problem.
UR - http://www.scopus.com/inward/record.url?scp=0001832837&partnerID=8YFLogxK
U2 - 10.4064/fm-159-1-1-50
DO - 10.4064/fm-159-1-1-50
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AN - SCOPUS:0001832837
SN - 0016-2736
VL - 159
SP - 1
EP - 50
JO - Fundamenta Mathematicae
JF - Fundamenta Mathematicae
IS - 1
ER -