Using almost disjoint coding, we show that for each $1<M<N<\omega$
consistently $\mathfrak{d}=\mathfrak{a}_g=\aleph_M<\mathfrak{c}=\aleph_N$,
where $\mathfrak{a}_g=\aleph_M$ is witnessed by a $\Pi^1_2$ maximal cofinitary
group.Comment: 22 page

Fischer, Vera

Friedman, Sy David

Schrittesser, David

Törnquist, Asger

English

arXiv

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.Comment: The exposition has been thoroughly revised; 33 page

Definable maximal cofinitary groups of intermediate size

Abstract

Similar works

Full text

Available Versions

arXiv.org e-Print Archive