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arXiv · 1802.10125

Guessing models and the approachability ideal

Rahman Mohammadpour·Boban Velickovic
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Abstract

Starting with two supercompact cardinals we produce a generic extension of the universe in which a principle that we call ${\rm GM}^+(ω_3,ω_1)$ holds. This principle implies ${\rm ISP}(ω_2)$ and ${\rm ISP}(ω_3)$, and hence the tree property at $ω_2$ and $ω_3$, the Singular Cardinal Hypothesis, and the failure of the weak square principle $\square(ω_2,λ)$, for all regular $λ\geq ω_2$. In addition, it implies that the restriction of the approachability ideal $I[ω_2]$ to the set of ordinals of cofinality $ω_1$ is the non stationary ideal on this set. The consistency of this last statement was previously shown by Mitchell.

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Rahman Mohammadpour, Boban Velickovic. 2019-05-18. Guessing models and the approachability ideal. https://arxiv.org/abs/1802.10125

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