Good projective witnesses

Vera Fischer; Sy David Friedman; David Schrittesser; Asger Törnquist

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arXiv · 1904.05823

Good projective witnesses

Vera Fischer·Sy David Friedman·David Schrittesser·Asger Törnquist

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Abstract

We develop a new forcing notion for adjoining self-coding cofinitary permutations and use it to show that consistently, the minimal cardinality $\mathfrak a_{\text{g}}$ of a maximal cofinitary group (MCG) is strictly between $\aleph_1$ and $\mathfrak{c}$, and there is a $\Pi^1_2$-definable MCG of this cardinality. Here $\Pi^1_2$ is optimal, making this result a natural counterpart to the Borel MCG of Horowitz and Shelah. Our theorem has its analogue in the realm of maximal almost disjoint (MAD) families, extending a line of results regarding the definability properties of MAD families in models with large continuum.

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BibTeX RIS

Vera Fischer, Sy David Friedman, David Schrittesser, Asger Törnquist. 2019-04-11. Good projective witnesses. https://arxiv.org/abs/1904.05823

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