Derived topologies on ordinals and stationary reflection

Abstract

We study the transfinite sequence of topologies on the ordinal numbers that is obtained through successive closure under Cantor's derivative operator on sets of ordinals, starting form the usual interval topology. We characterize the non-isolated points in the ξ-th topology as those ordinals that satisfy a strong iterated form of stationary reflection, which we call ξ-simultaneous-reflection. We prove some properties of the ideals of non-ξ-simultaneous-stationary sets and identify their tight connection with indescribable cardinals. We then introduce a new natural notion of Π 1 ξ-indescribability, for any ordinal ξ, which extends to the transfinite the usual notion of Π 1 n-indescribability, and prove that in the constructible universe L, a regular cardinal is (ξ + 1)-simultaneouslyreflecting if and only if it is Π 1 ξ-indescribable, a result that generalizes to all ordinals ξ previous results of Jensen [28] in the case ξ = 2, and Bagaria-Magidor-Sakai [5] in the case ξ = n. This yields a complete characterization in L of the non-discreteness of the ξ-topologies, both in terms of iterated stationary reflection and in terms of indescribability. 1. Introduction Cantor's derivative operator on a topological space (X, τ) is the map d τ that assigns to every subset A of X the set of its limit points. By declaring the sets d τ (A) to be open one generates a finer topology. Through successive applications of this process, and by taking unions at limit stages, one obtains a polytopological space (X, τ 0 , τ 1 ,. .. , τ ξ ,. . .), where τ 0 = τ and τ ζ ⊆ τ ξ whenever ζ < ξ. The study of such spaces has been of great interest in recent years, not so much in topology but, perhaps surprisingly, in proof theory and modal logic. When (X, τ) is scattered, the derived polytopological spaces (X, τ 0 , τ 1 ,. .. , τ ξ ,. . .) are known in the literature as GLP-spaces, for they provide a natural topological interpretation of the logic GLP (Japaridze [24]), namely the polymodal extension of the classical Gödel-Löb provability logic GL to infinitely-many modal operators [n],

A Refinement of the Ramsey Hierarchy via Indescribability

“…We say that κ > is a Ramsey cardinal if for every function f : [κ] < → 2 there is a set H ⊆ κ of size κ which is homogeneous for f, meaning that f ↾ [H] n is constant for all n < . 1 The study of Ramsey-like properties of uncountable cardinals has been a central concern of set theorists working on large cardinals and infinitary combinatorics, with renewed interest in recent years (see [3, 6, 7, 12, 15-17, 19, 22, 23, 25]). In this article, we study Ramsey-like properties of uncountable cardinals in which homogeneous sets are demanded to have degrees of indescribability: for example, a cardinal κ is 1-Π 1 n -Ramsey where n < if and only if every function f : [κ] < → 2 has a Π 1 n -indescribable homogeneous set H ⊆ κ.…”

“…1 The study of Ramsey-like properties of uncountable cardinals has been a central concern of set theorists working on large cardinals and infinitary combinatorics, with renewed interest in recent years (see [3, 6, 7, 12, 15-17, 19, 22, 23, 25]). In this article, we study Ramsey-like properties of uncountable cardinals in which homogeneous sets are demanded to have degrees of indescribability: for example, a cardinal κ is 1-Π 1 n -Ramsey where n < if and only if every function f : [κ] < → 2 has a Π 1 n -indescribable homogeneous set H ⊆ κ. Among other things, we show that hypotheses of this kind and their generalizations lead to a strict refinement of Feng's [12] original Ramsey hierarchy: we isolate large cardinal hypotheses which provide strictly increasing hierarchies between Feng's Π α -Ramsey and Π α+1 -Ramsey cardinals for all odd α < and for all ≤ α < κ (see Figure 4).…”

“…2 This leads naturally to the consideration of large cardinal ideals: for example, Baumgartner showed that if κ is a Ramsey cardinal then the collection of non-Ramsey subsets of κ is a nontrivial normal ideal on κ called the Ramsey ideal. Similarly, a set S ⊆ κ is Π 1 n -indescribable if for all A ⊆ V κ and all Π 1 n sentences ϕ, if (V κ , ∈, A) |= ϕ then there is an α ∈ S such that (V α , ∈, A ∩ V α ) |= ϕ, and the collection Π 1 n (κ) of non-Π 1 n -indescribable subsets of κ is a normal ideal on κ when κ is Π 1 n -indescribable. Baumgartner proved that a cardinal κ is Ramsey if and only if κ is pre-Ramsey, 3 κ is Π 1 1 -indescribable and the union of the Π 1 1 -indescribability ideal on κ and the pre-Ramsey ideal on κ generate a nontrivial ideal 4 which equals the Ramsey ideal; furthermore, reference to these ideals cannot be removed from this characterization because the least cardinal which is both pre-Ramsey and Π 1 1indescribable is not Ramsey.…”

“…Sharpe and Welch [25,Definition 3.21] extended the notion of Π 1 n -indescribability of a cardinal κ where n < to that of Π 1 -indescribability where < κ + by demanding that the existence of a winning strategy for a particular player in a certain finite game played at κ implies that the same player has a winning strategy in the analogous game played at some cardinal less than κ. Later, Bagaria [1,Definition 4.2] gave an alternative definition of the Π 1 -indescribability of a cardinal κ for < κ using the indescribability of rank-initial segments of the set-theoretic universe by certain sentences in an infinitary logic. In what follows we will use Bagaria's definition since it seems easier to work with in this context.…”

“…In what follows we will use Bagaria's definition since it seems easier to work with in this context. Bagaria extended the definitions of the classes of Π 1 n and Σ 1 n formulas to define the natural classes of Π 1 and Σ 1 formulas for all ordinals . For example, a formula is Π 1 if it is of the form n< ϕ n where each ϕ n is Π 1 n and it contains finitely-many free second order variables.…”