Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-28T08:27:41.034Z Has data issue: false hasContentIssue false
  • English
  • Français

ℙmax variations for separating club guessing principles

Published online by Cambridge University Press:  12 March 2014

Tetsuya Ishiu  and
Paul B. Larson
Tetsuya Ishiu
Affiliation:
Department of Mathematics, Miami University, Oxford, Ohio 45056, USA, E-mail: ishiut@muohio.edu
Paul B. Larson
Affiliation:
Department of Mathematics, Miami University, Oxford, Ohio 45056, USA, E-mail: larsonpb@muohio.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

In his book on ℙmax [7], Woodin presents a collection of partial orders whose extensions satisfy strong club guessing principles on ω1. 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 ω1. It was shown in [1] that the Continuum Hypothesis does not imply (+) and that (+) does not imply the existence of a club guessing sequence ω1. 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.

Keywords

club guessingcondensationℙmax
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 2 , June 2012 , pp. 532 - 544
Copyright
Copyright © Association for Symbolic Logic 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Ishiu, T. and Larson, P., Some results about (+) proved by iterated forcing, this Journal, vol. 77 (2012). no. 2, pp. 515–531.Google Scholar
[2] Kunen, K., A combinatorial principle consistent with MA + ̚CH. Circulated note. 02 1971.Google Scholar
[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.Google Scholar
[4] Larson, P., The canonical function game. Archive for Mathematical Logic, vol. 44 (2005). no. 7, pp. 817–827.CrossRefGoogle Scholar
[5] Larson, P., Forcing over models of determinacy, Handbook of set theory (Kanmori, A. and Foreman, M., editors). Springer, 2010, pp. 2121–2177.Google Scholar
[6] Shelah, S. and Zapletal, J., Canonical models for ℵ-combinatorics, Annals of Pure and Applied Logic, vol. 98 (1999), no. 1–3, pp. 217–259.Google Scholar
[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.Google Scholar