Borel sets with large squares

Saharon Shelah

Fundamenta Mathematicae (1999)

  • Volume: 159, Issue: 1, page 1-50
  • ISSN: 0016-2736

Abstract

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 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 2 ℵ 0 -square and even a perfect square, and also for ⊗ μ ' if ψ ∈ L ω 1 , ω has a model of cardinality μ then it has a model of cardinality continuum generated in a “nice”, “absolute” way. Assuming M A + 2 ℵ 0 > μ for transparency, those three conditions ( ⊕ μ , ⊗ μ and ⊗ μ ' ) are equivalent, and from this we deduce that e.g. ∧ α < ω 1 [ 2 ℵ 0 ≥ ℵ α ⇒ ￢ ⊗ ℵ α ] , and also that m i n μ : ⊗ μ , if < 2 ℵ 0 , has cofinality ℵ 1 .   We also deal with Borel rectangles and related model-theoretic problems.

How to cite

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Shelah, Saharon. "Borel sets with large squares." Fundamenta Mathematicae 159.1 (1999): 1-50. <http://eudml.org/doc/212318>.

@article{Shelah1999,
abstract = { 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 $2^\{ℵ_0\}$-square and even a perfect square, and also for $⊗^\{\prime \}_μ$ if $ψ ∈ L_\{ω_1, ω\}$ has a model of cardinality μ then it has a model of cardinality continuum generated in a “nice”, “absolute” way. Assuming $MA + 2^\{ℵ_0\} > μ$ for transparency, those three conditions ($⊕_μ$, $⊗_μ$ and $⊗^\{\prime \}_μ$) are equivalent, and from this we deduce that e.g. $∧_\{α < ω_1\}[ 2^\{ℵ_0\}≥ ℵ_α ⇒ ￢ ⊗_\{ℵ_α\}]$, and also that $min\{μ: ⊗_μ\}$, if $ < 2^\{ℵ_0\}$, has cofinality $ℵ_1$.   We also deal with Borel rectangles and related model-theoretic problems.},
author = {Shelah, Saharon},
journal = {Fundamenta Mathematicae},
keywords = {Borel set; perfect square; Cohen reals; analytic set; Hanf numbers; rectangle containment; Martin's Axiom; Borel rectangles},
language = {eng},
number = {1},
pages = {1-50},
title = {Borel sets with large squares},
url = {http://eudml.org/doc/212318},
volume = {159},
year = {1999},
}

TY - JOUR
AU - Shelah, Saharon
TI - Borel sets with large squares
JO - Fundamenta Mathematicae
PY - 1999
VL - 159
IS - 1
SP - 1
EP - 50
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 $2^{ℵ_0}$-square and even a perfect square, and also for $⊗^{\prime }_μ$ if $ψ ∈ L_{ω_1, ω}$ has a model of cardinality μ then it has a model of cardinality continuum generated in a “nice”, “absolute” way. Assuming $MA + 2^{ℵ_0} > μ$ for transparency, those three conditions ($⊕_μ$, $⊗_μ$ and $⊗^{\prime }_μ$) are equivalent, and from this we deduce that e.g. $∧_{α < ω_1}[ 2^{ℵ_0}≥ ℵ_α ⇒ ￢ ⊗_{ℵ_α}]$, and also that $min{μ: ⊗_μ}$, if $ < 2^{ℵ_0}$, has cofinality $ℵ_1$.   We also deal with Borel rectangles and related model-theoretic problems.
LA - eng
KW - Borel set; perfect square; Cohen reals; analytic set; Hanf numbers; rectangle containment; Martin's Axiom; Borel rectangles
UR - http://eudml.org/doc/212318
ER -

References

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  11. [Sh 8] S. Shelah, Two cardinal compactness, Israel J. Math. 9 (1971), 193-198; see also Notices Amer. Math. Soc. 18 (1971), 425. 
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  13. [Sh 37] S. Shelah, A two-cardinal theorem, Proc. Amer. Math. Soc. 48 (1975), 207-213. 
  14. [Sh 49] S. Shelah, A two-cardinal theorem and a combinatorial theorem, ibid. 62 (1976), 134-136. 
  15. [Sh 202] S. Shelah, On co-κ-Suslin relations, Israel J. Math. 47 (1984), 139-153. 
  16. [Sh 262] S. Shelah, The number of pairwise non-elementarily-embeddable models, J. Symbolic Logic 54 (1989), 1431-1455. Zbl0701.03015
  17. [Sh 288] S. Shelah, Strong partition relations below the power set: consistency, was Sierpiński right? II, in: Sets, Graphs and Numbers (Budapest, 1991), Colloq. Math. Soc. János Bolyai 60, North-Holland, 1992, 637-638. 
  18. [Sh 532] S. Shelah, More on co-κ-Suslin equivalence relations, in preparation. 

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