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On partitions into stationary sets

  • Karel Prikry (a1) and Robert M. Solovay (a2)

We shall apply some of the results of Jensen [4] to deduce new combinatorial consequences of the axiom of constructibility, V = L. We shall show, among other things, that if V = L then for each cardinal λ there is a set A ⊆ λ such that neither A nor λ – A contain a closed set of type ω1. This is an extension of a result of Silver who proved it for λ = ω2, providing a partial answer to Problem 68 of Friedman [2].

The main results of this paper were obtained independently by both authors.

If λ is an ordinal, E is said to be Mahlo (or stationary) in λ, if λ – E does not contain a closed cofinal subset of λ.

Consider the statements:

(J1) There is a class E of limit ordinals and a sequence Cλ defined on singular limit ordinals λ such that

(i) E ⋂ μ is Mahlo in μ for all regular > ω;

(ii) Cλ is closed and unbounded in λ;

(iii) if γ < is a limit point of Cλ, then γ is singular, γ ∉ E and Cγ = γ ⋂ Cλ.

For each infinite cardinal κ:

(J2,κ) There is a set E ⊂ κ+ and a sequence Cλ(Lim(λ), λ < κ+) such that

(i) E is Mahlo in κ+;

(ii) Cλ is closed and unbounded in λ;

(iii) if cf(λ) < κ, then card Cλ < κ;

(iv) if γ < λ is a limit point of Cλ then γ ∉ E and Cγ = ϣ ⋂ Cλ.

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[1]Friedman, H., On closed sets of ordinals, Proceedings of the American Mathematical Society, vol. 43 (1974), pp. 190192.
[2]Friedman, H., Ninety-four problems in mathematical logic (to appear).
[3]Jech, T. J., Some combinatorial problems concerning uncountable cardinals, Annals of Mathematical Logic, vol. 5 (1973), pp. 165198.
[4]Jensen, R. B., The fine structure of the constructible hierarchy, Annals of Mathematical Logic, vol. 4 (1972), pp. 229308.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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