Journal of Symbolic Logic
- J. Symbolic Logic
- Volume 76, Issue 4 (2011), 1441-1452.
MRP, tree properties and square principles
We show that MRP+MA implies that ITP(λ,ω2) holds for all cardinal λ ≥ ω2. This generalizes a result by Weiß 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.
J. Symbolic Logic, Volume 76, Issue 4 (2011), 1441-1452.
First available in Project Euclid: 11 October 2011
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 03E35: Consistency and independence results 03E50: Continuum hypothesis and Martin's axiom [See also 03E57] 03E55: Large cardinals 03E57: Generic absoluteness and forcing axioms [See also 03E50]
Strullu, Remi. MRP , tree properties and square principles. J. Symbolic Logic 76 (2011), no. 4, 1441--1452. doi:10.2178/jsl/1318338859. https://projecteuclid.org/euclid.jsl/1318338859