Large cardinal ideals

Abstract

Building on work of Holy, Lücke and Njegomir [30] on small embedding characterizations of large cardinals, we use some classical results of Baumgartner (see [7] and [9]), to give characterizations of several well-known large cardinal ideals, including the Ramsey ideal, in terms of generic elementary embeddings; we also point out some seemingly inherent differences between small embedding and generic embedding characterizations of subtle cardinals. Additionally, we present a simple and uniform proof which shows that, when κ is weakly compact, many large cardinal ideals on κ are nowhere κ-saturated. Lastly, we survey some recent consistency results concerning the weakly compact ideal as well as some recent results on the subtle, ineffable and Π 1 1 -indescribable ideals on Pκλ, and we close with a list of open questions. Contents1. Introduction 2. Preliminaries 2.1. Basic terminology and facts about ideals 2.2. Small embeddings 2.3. Embedding characterizations of stationarity 3. Large cardinal ideals and elementary embeddings 3.1. The Π m n -indescribability ideals 3.2. The subtle ideal 3.3. The almost ineffable and ineffable ideals 3.4. The Ramsey ideal 4. Splitting positive sets assuming weak compactness 5. Consistency results 5.1. n-clubs and indescribability embeddings 5.2. A theorem of Hellsten 5.3. The weakly compact reflection principle 5.4. A (κ)-like principle consistent with weak compactness 6. Large cardinal ideals on P κ λ 6.1. Stationary vs. strongly stationary subsets of P κ λ 6.2. Indescribable subsets of P κ λ 6.3. Subtle, strongly subtle and ineffable subsets of P κ λ 6.4. Generic embedding characterizations of large cardinal ideals on P κ λ 7. Questions