Borel sets with large squares

Shelah, Saharon

Fundamenta Mathematicae, Tome 159 (1999), p. 1-50 / Harvested from The Polish Digital Mathematics Library

Résumé

 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.

Détail

Publié le : 1999-01-01

Zbl 0941.03052

EUDML-ID : urn:eudml:doc:212318

Bibliographie

[00000] [CK] C. C. Chang and J. H. Keisler, Model Theory, Stud. Logic Found. Math. 73, North-Holland, Amsterdam, 1973.

[00001] [EHMR] P. Erdős, A. Hajnal, A. Máté and R. Rado, Combinatorial Set Theory: Partition Relations for Cardinals, Stud. Logic Found. Math. 106, North-Holland, Amsterdam, 1984. | Zbl 0573.03019

[00002] [GcSh 491] M. Gilchrist and S. Shelah, Identities on cardinals less than $ℵ_ω$, J. Symbolic Logic 61 (1996), 780-787. | Zbl 0866.03026

[00003] [HrSh 152] L. Harrington and S. Shelah, Counting equivalence classes for co-κ-Suslin equivalence relations, in: Logic Colloquium '80 (Prague, 1980), D. van Dalen et al. (eds.), Stud. Logic Found. Math. 108, North-Holland, Amsterdam, 1982, 147-152.

[00004] [Ke71] H. J. Keisler, Model Theory for Infinitary Logic. Logic with Countable Conjunctions and Finite Quantifiers, Stud. Logic Found. Math. 62, North-Holland, Amsterdam, 1971.

[00005] [Mo] M. Morley, Omitting classes of elements, in: The Theory of Models, North-Holland, 1965, 265-273.

[00006] [Sh a] S. Shelah, Classification Theory and the Number of Nonisomorphic Models, Stud. Logic Found. Math. 92, North-Holland, Amsterdam, xvi+544 pp., 1978.

[00007] [Sh c] S. Shelah, Classification Theory and the Number of Nonisomorphic Models, Stud. Logic Found. Math. 92, North-Holland, Amsterdam, xxxiv+705 pp., 1990.

[00008] [Sh e] S. Shelah, Non-Structure Theory, Oxford Univ. Press, in preparation.

[00009] [Sh g] S. Shelah, Cardinal Arithmetic, Oxford Logic Guides 29, Oxford Univ. Press, 1994.

[00010] [Sh 8] S. Shelah, Two cardinal compactness, Israel J. Math. 9 (1971), 193-198; see also Notices Amer. Math. Soc. 18 (1971), 425.

[00011] [Sh 18] S. Shelah, On models with power-like orderings, J. Symbolic Logic 37 (1972), 247-267. | Zbl 0273.02036

[00012] [Sh 37] S. Shelah, A two-cardinal theorem, Proc. Amer. Math. Soc. 48 (1975), 207-213.

[00013] [Sh 49] S. Shelah, A two-cardinal theorem and a combinatorial theorem, ibid. 62 (1976), 134-136.

[00014] [Sh 202] S. Shelah, On co-κ-Suslin relations, Israel J. Math. 47 (1984), 139-153.

[00015] [Sh 262] S. Shelah, The number of pairwise non-elementarily-embeddable models, J. Symbolic Logic 54 (1989), 1431-1455. | Zbl 0701.03015

[00016] [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.

[00017] [Sh 532] S. Shelah, More on co-κ-Suslin equivalence relations, in preparation.