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MRP, tree properties and square principles

Published online by Cambridge University Press:  12 March 2014

Remi Strullu
Remi Strullu*
Affiliation:
Equipe de Logique Mathématique, Université Paris Diderot Paris 7,UFR de Mathématiques Case 7012, Site Chevaleret, 75205 Paris Cedex 13, France, E-mail: rstrullu@logique.jussieu.fr
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Abstract

We show that MRP + MA implies that ITP(λ,ω2) holds for all cardinal λ ≥ ω2. This generalizes a result by Weiβ who showed that PFA implies that ITP(λ, ω2) holds for all cardinal λ ≥ ω2. Consequently any of the known methods to prove MRP + MA consistent relative to some large cardinal hypothesis requires the existence of a strongly compact cardinal. Moreover if one wants to force MRP + MA with a proper forcing, it requires at least a supercompact cardinal. We also study the relationship between MRP and some weak versions of square. We show that MRP implies the failure of □(λ, ω) for all λ ≥ ω2 and we give a direct proof that MRP + MA implies the failure of □(λ, ω1) for all λ ≥ ω2.

Keywords

MRPITPsquare sequencestrongly compact cardinalconsistency results
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 76 , Issue 4 , December 2011 , pp. 1441 - 1452
Copyright
Copyright © Association for Symbolic Logic 2011

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References

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