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2 results

  • Possible behaviours of the reflection ordering of stationary sets

  • Jiří Witzany
  • Journal:
    The Journal of Symbolic Logic / Volume 60 / Issue 2 / June 1995
    Published online by Cambridge University Press:
    12 March 2014, pp. 534-547
    Print publication:
    June 1995

  • If S, T are stationary subsets of a regular uncountable cardinal κ, we say that S reflects fully in T, S < T, if for almost all α ∈ T (except a nonstationary set) S ∩ α stationary in α. This relation is known to be a well-founded partial ordering. We say that a given poset P is realized by the reflection ordering if there is a maximal antichain 〈Xp: p ∈ P〉 of stationary subsets of Reg(κ) so that

    We prove that if , and P is an arbitrary well-founded poset of cardinality ≤ κ+ then there is a generic extension where P is realized by the reflection ordering on κ.

  • Full reflection at a measurable cardinal

  • Thomas Jech, Jiří Witzany
  • Journal:
    The Journal of Symbolic Logic / Volume 59 / Issue 2 / June 1994
    Published online by Cambridge University Press:
    12 March 2014, pp. 615-630
    Print publication:
    June 1994

  • A stationary subset S of a regular uncountable cardinal κreflects fully at regular cardinals if for every stationary set T ⊆ κ of higher order consisting of regular cardinals there exists an α Є T such that S ∩ α is a stationary subset of α. Full Reflection states that every stationary set reflects fully at regular cardinals. We will prove that under a slightly weaker assumption than κ having the Mitchell order κ++ it is consistent that Full Reflection holds at every λ ≤ κ and κ is measurable.