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max variations for separating club guessing principles

  • Tetsuya Ishiu (a1) and Paul B. Larson (a2)

Abstract

In his book on ℙmax [7], Woodin presents a collection of partial orders whose extensions satisfy strong club guessing principles on ω 1. In this paper we employ one of the techniques from this book to produce ℙmax variations which separate various club guessing principles. The principle (+) and its variants are weak guessing principles which were first considered by the second author [4] while studying games of length ω 1. It was shown in [1] that the Continuum Hypothesis does not imply (+) and that (+) does not imply the existence of a club guessing sequence ω 1. In this paper we give an alternate proof of the second of these results, using Woodin's ℙmax technology, showing that a strengthening of (+) does not imply a weakening of club guessing known as the Interval Hitting Principle. The main technique in this paper, in addition to the standard ℙmax machinery, is the use of condensation principles to build suitable iterations.

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References

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[1] Ishiu, T. and Larson, P., Some results about (+) proved by iterated forcing, this Journal, vol. 77 (2012). no. 2, pp. 515531.
[2] Kunen, K., A combinatorial principle consistent with MA + ̚CH. Circulated note. 02 1971.
[3] Larson, P., The stationary tower. University Lecture Series, vol. 32, American Mathematical Society, Providence, RI, 2004. Notes on a course by William Hugh Woodin.
[4] Larson, P., The canonical function game. Archive for Mathematical Logic, vol. 44 (2005). no. 7, pp. 817827.
[5] Larson, P., Forcing over models of determinacy, Handbook of set theory (Kanmori, A. and Foreman, M., editors). Springer, 2010, pp. 21212177.
[6] Shelah, S. and Zapletal, J., Canonical models for ℵ-combinatorics, Annals of Pure and Applied Logic, vol. 98 (1999), no. 1–3, pp. 217259.
[7] Woodin, W.H., The axiom of determinacy, forcing axioms, and the nonstationary ideal, second revised ed., de Gruyter Series in Logic and its Applications, vol. 1. Walter de Gruyter & Co., Berlin. 2010.
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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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