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Ramsey-like cardinals II

  • Victoria Gitman (a1) and P. D. Welch (a2)


This paper continues the study of the Ramsey-like large cardinals introduced in [5] and [14]. 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 αω1, that they are downward absolute to L for , 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 [5] 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 [14] as an upper bound on the consistency strength of the Intermediate Chang's Conjecture.



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[1]Beller, A., Jensen, R., and Welch, P., Coding the universe, London Mathematical Society Lecture Note Series, vol. 47, Cambridge University Press, Cambridge, 1982.
[2]Donder, Hans-Dieter and Levinski, Jean-Pierre, Some principles related to Chang's conjecture, Annals of Pure and Applied Logic, vol. 45 (1989), no. 1, pp. 39101.
[3]Feng, Q., A hierarchy of Ramsey cardinals, Annals of Pure and Applied Logic, vol. 49 (1990), no. 3, pp. 257277.
[4]Gaifman, H., Elementary embeddings of models of set-theory and certain subtheories, Axiomatic set theory (Proceedings of the Symposium on Pure Mathematics, vol. XIII, part II, University of California, Los Angeles, California, 1967), American Mathematical Society, Providence, R.I., 1974, pp. 33101.
[5]Gitman, V., Ramsey-like cardinals, this Journal, vol. 76 (2011), no. 2, pp. 519540.
[6]Gitman, V. and Johnstone, T., Indestructibility for Ramsey-like cardinals, in preparation, 2010.
[7]Hamkins, J. D., Forcing and large cardinals, manuscript, 2007.
[8]Kanamori, A., The higher infinite, second ed., Springer Monographs in Mathematics, Springer–Verlag, New York, 2003.
[9]Kunen, K., Some applications of iterated ultrapowers in set theory, Annals of Pure and Applied Logic, vol. 1 (1970), pp. 179227.
[10]Kunen, K., Saturated ideals, this Journal, vol. 43 (1978), no. 1, pp. 6576.
[11]Mitchell, W. J., Ramsey cardinals and constructibility, this Journal, vol. 44 (1979), no. 2, pp. 260266.
[12]Schindler, R., Semi-proper forcing, remarkable cardinals, and Bounded Martin's Maximum, Mathematical Logic Quarterly, vol. 50 (2004), no. 6, pp. 527532.
[13]Villaveces, A., Chains of end elementary extensions of models of set theory, this Journal, vol. 63 (1998), no. 3, pp. 11161136.
[14]Welch, P. D. and Sharpe, I., Greatly Erdős cardinals and some generalizations to the Chang and Ramsey properties, Annals of Pure and Applied Logic, to appear.
[15]Zeman, M., Inner models and large cardinals, de Gruyter Series in Logic and its Applications, vol. 5, Walter de Gruyter & Co., Berlin, 2002.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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