Ideal Operators and Higher Indescribability

Abstract

We investigate properties of the ineffability and the Ramsey operator, and a common generalization of those that was introduced by the second author, with respect to higher indescribability, as introduced by the first author. This extends earlier investigations on the ineffability operator by James Baumgartner, and on the Ramsey operator by Qi Feng, by Philip Welch et al., and by the first author.

Higher indescribability and derived topologies

“…Abe [1,Lemma 4.1] showed that if P κ X is Π 1 n -indescribable then Π 1 n (κ, X) is a strongly normal ideal on P κ X. As pointed out in [16], a straightforward application of the arguments for [1, Lemma 4.1] and [2,Proposition 4.4], which is left to the reader, establishes the following.…”

“…The classical notion of Π m n -indescribability studied by Levy [22] was generalized to the two-cardinal setting in a set of handwritten notes by Baumgartner (see [9,Section 4]). More recently, various transfinite generalizations of classical Π 1 n -indescribability, involving certain infinitary formulas have been studied in the cardinal context [2,3,4,11,13,15] and in the two-cardinal context [12].…”

“…Additionally, in Section 4, we answer several questions asked by the first author and Peter Holy [15] and the first author and Philip White [16] concerning the relationship between Ramseyness and indescribability. For example, answering [15,Question 10.9] in the affirmative, we show that the existence of a 2-Ramsey cardinal is strictly stronger in consistency strength than the existence of a 1-Π 1 1 -Ramsey cardinal.…”

“…Additionally, in Section 4, we answer several questions asked by the first author and Peter Holy [15] and the first author and Philip White [16] concerning the relationship between Ramseyness and indescribability. For example, answering [15,Question 10.9] in the affirmative, we show that the existence of a 2-Ramsey cardinal is strictly stronger in consistency strength than the existence of a 1-Π 1 1 -Ramsey cardinal. In other words, the existence of an uncountable cardinal κ such that for every regressive function f : [κ] <ω → κ there is a set H ⊆ κ which is positive for the Ramsey ideal homogeneous for f , is strictly stronger in consistency strength than the existence of a cardinal κ such that for every regressive function f : [κ] <ω → κ there is a set H ⊆ κ that is positive for the Π 1 1 -indescribability ideal and homogeneous for f .…”

Ramsey-like operators

“…By similar means to Theorem 14.1, many of the results from [6] for the Ramsey operator can be extended to our generalized operators in a fairly straightforward way. Amongst other things, these further generalizations are planned to be included in a follow-up paper [7]. Let us present one final easy sample result here, namely that applications of our operators to different indescribability ideals give rise to a proper hierarchy of large cardinal notions (this is a weak generalized analogue of results from [6]).…”