2011/04/22 by Victoria Gitman, Gitman, Victoria, Philip Welch +1
Computer Science · Mathematics · #03E35 #03E55 #Advanced Topology and Set Theory #Computability, Logic, AI Algorithms #FOS: Mathematics #Limits and Structures in Graph Theory #Logic (math.LO) #math.LO #msc:03E35 #msc:03E55
paper · pdf · doi:10.48550/arxiv.1104.4448
arxiv created 2011/04/22 · openalex publication_date 2011/04/22 · arxiv updated 2011/04/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This paper continues the study of the Ramsey-like large cardinals. Ramsey-like cardinals are defined by generalizing the characterization of Ramsey cardinals via the existence of elementary embeddings. Ultrafilters derived from such embeddings are fully iterable and so it is natural to ask about large cardinal notions asserting the existence of ultrafilters allowing only α-many iterations for some countable ordinal α. Here we study such α-iterable cardinals. We show that the α-iterable cardinals form a strict hierarchy for α≤ω1, that they are downward absolute to L for α<ω1L, and that the consistency strength of Schindler's remarkable cardinals is strictly between 1-iterable and 2-iterable cardinals.