## Ramsey-like cardinals. II.(English)Zbl 1222.03055

Summary: This paper continues the study of the Ramsey-like large cardinals introduced in Part I [V. Gitman, ibid. 76, No. 2, 519–540 (2011; Zbl 1222.03054)] and in [I. Sharpe and P. D. Welch, Ann. Pure Appl. Logic 162, No. 11, 863–902 (2011; Zbl 1270.03071)]. 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 $$\alpha$$-many iterations for some countable ordinal $$\alpha$$. Here we study such $$\alpha$$-iterable cardinals. We show that the $$\alpha$$-iterable cardinals form a strict hierarchy for $$\alpha \leq \omega _{1}$$, that they are downward absolute to $$L$$ for $$\alpha <\omega _{1}^{L}$$, and that the consistency strength of Schindler’s remarkable cardinals is strictly between 1-iterable and 2-iterable cardinals.
We show that the strongly Ramsey and super Ramsey cardinals from Part I are downward absolute to the core model $$K$$. Finally, we use a forcing argument from a strongly Ramsey cardinal to separate the notions of Ramsey and virtually Ramsey cardinals. These were introduced in [Sharpe and Welch, loc. cit.] as an upper bound on the consistency strength of the Intermediate Chang’s Conjecture.

### MSC:

 3e+55 Large cardinals

### Citations:

Zbl 1222.03054; Zbl 1270.03071
Full Text:

### References:

 [1] Axiomatic set theory (Proceedings of the Symposium on Pure Mathematics, vol. XIII, part II, University of California, Los Angeles, California, 1967) pp 33– (1974) [2] DOI: 10.1016/0168-0072(90)90028-Z · Zbl 0719.03025 [3] DOI: 10.1016/0168-0072(89)90030-4 · Zbl 0714.03048 [4] Coding the universe 47 (1982) · Zbl 0468.03031 [5] Inner models and large cardinals 5 (2002) [6] Annals of Pure and Applied Logic [7] Ramsey-like cardinals 76 pp 519– (2011) · Zbl 1222.03054 [8] DOI: 10.1002/malq.200410002 · Zbl 1058.03052 [9] Ramsey cardinals and constructibility 44 pp 260– (1979) [10] Annals of Pure and Applied Logic 1 pp 179– (1970) [11] The higher infinite (2003) [12] Saturated ideals 43 pp 65– (1978) [13] Chains of end elementary extensions of models of set theory 63 pp 1116– (1998) · Zbl 0915.03034
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.