Borel sets with large squares

• Autorzy: Saharon Shelah
• Języki publikacji: EN

Abstrakty

EN

 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 $MA + 2^{ℵ_0} > μ$ 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
$min{μ: ⊗_μ}$, if $ < 2^{ℵ_0}$, has cofinality $ℵ_1$.
  We also deal with Borel rectangles and related model-theoretic problems.

Kategorie tematyczne

• 03E05: Other combinatorial set theory
• 03E15: Descriptive set theory
• 03E35: Consistency and independence results
• 03C55: Set-theoretic model theory

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1999
• Tom: 159
• Numer: 1
• Strony: 1-50

Daty

wydano

1999

otrzymano

1994-02-28

poprawiono

1996-12-23

poprawiono

1998-03-11

Twórcy

autor

Saharon Shelah

• Institute of Mathematics, The Hebrew University, Jerusalem 91904, Israel
• Mathematics Department, Rutgers University, New Brunswick, New Jersey 08854, U.S.A.

Bibliografia

• [CK] C. C. Chang and J. H. Keisler, Model Theory, Stud. Logic Found. Math. 73, North-Holland, Amsterdam, 1973.
• [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.
• [GcSh 491] M. Gilchrist and S. Shelah, Identities on cardinals less than $ℵ_ω$, J. Symbolic Logic 61 (1996), 780-787.
• [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.
• [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.
• [Mo] M. Morley, Omitting classes of elements, in: The Theory of Models, North-Holland, 1965, 265-273.
• [Sh a] S. Shelah, Classification Theory and the Number of Nonisomorphic Models, Stud. Logic Found. Math. 92, North-Holland, Amsterdam, xvi+544 pp., 1978.
• [Sh c] S. Shelah, Classification Theory and the Number of Nonisomorphic Models, Stud. Logic Found. Math. 92, North-Holland, Amsterdam, xxxiv+705 pp., 1990.
• [Sh e] S. Shelah, Non-Structure Theory, Oxford Univ. Press, in preparation.
• [Sh g] S. Shelah, Cardinal Arithmetic, Oxford Logic Guides 29, Oxford Univ. Press, 1994.
• [Sh 8] S. Shelah, Two cardinal compactness, Israel J. Math. 9 (1971), 193-198; see also Notices Amer. Math. Soc. 18 (1971), 425.
• [Sh 18] S. Shelah, On models with power-like orderings, J. Symbolic Logic 37 (1972), 247-267.
• [Sh 37] S. Shelah, A two-cardinal theorem, Proc. Amer. Math. Soc. 48 (1975), 207-213.
• [Sh 49] S. Shelah, A two-cardinal theorem and a combinatorial theorem, ibid. 62 (1976), 134-136.
• [Sh 202] S. Shelah, On co-κ-Suslin relations, Israel J. Math. 47 (1984), 139-153.
• [Sh 262] S. Shelah, The number of pairwise non-elementarily-embeddable models, J. Symbolic Logic 54 (1989), 1431-1455.
• [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.
• [Sh 532] S. Shelah, More on co-κ-Suslin equivalence relations, in preparation.