Forcing Axioms and the Definabilty of the Nonstationary Ideal on ω₁

2022/08/10 by Hoffelner, Stefan, Larson, Paul, Schindler, Ralf +1 · 1 citation
#03E35 #03E45 #03E47 #03E55 #03E57 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2208.05288

Abstract

We show that under \BMM and "there exists a Woodin cardinal", the nonstationary ideal on ω₁ can not be defined by a Σ₁ formula with parameter A ⊂ ω₁. We show that the same conclusion holds under the assumption of Woodin's (∗)-axiom. We further show that there are universes where \BPFA holds and \NS is Σ₁(ω₁)-definable. Last we show that if the canonical inner model with one Woodin cardinal M₁ exists, there is a universe where \NS is saturated, Σ₁(ω₁)-definable and \MA holds.

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• NS_ω1 saturated, Δ₁ ( \ ω₁ \ )-definable and a Δ¹₄-definable well-order of the reals

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