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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: [email protected]
Paul B. Larson
Affiliation:
Department of Mathematics, Miami University, Oxford, Ohio 45056, USA, E-mail: [email protected]
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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

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References

REFERENCES

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