Projective well-orders and coanalytic witnesses

TL;DR

This paper develops a forcing method to construct models with definable well-orders of the reals and specific cardinal characteristics, preserving complex definable witnesses, advancing understanding of the interplay between set-theoretic properties and definability.

Contribution

Introduces a forcing notion that preserves definable witnesses for cardinal characteristics and well-orders, achieving optimal complexity results in a new model of set theory.

Findings

Constructed a model with ===\u00a71<2^{_0}=_2 with , ,  having _1 witnesses.

Established the consistency of a _3 well-order of the reals with =_2 and various inequalities among , , .

Developed methods to preserve only sufficiently definable witnesses, differing from previous approaches.

Abstract

We further develop a forcing notion known as Coding with Perfect Trees and show that this poset preserves, in a strong sense, definable $P$-points, definable tight MAD families and definable selective independent families. As a result, we obtain a model in which $\mathfrak{a}=\mathfrak{u}=\mathfrak{i}=\aleph_1<2^{\aleph_0}=\aleph_2$, each of $\mathfrak{a}$, $\mathfrak{u}$, $\mathfrak{i}$ has a $\Pi^1_1$ witness and there is a $\Delta^1_3$ well-order of the reals. Note that both the complexity of the witnesses of the above combinatorial cardinal characteristics, as well as the complexity of the well-order are optimal. In addition, we show that the existence of a $\Delta^1_3$ well-order of the reals is consistent with $\mathfrak{c}=\aleph_2$ and each of the following: $\mathfrak{a}=\mathfrak{u}<\mathfrak{i}$, $\mathfrak{a}=\mathfrak{i}<\mathfrak{u}$, $\mathfrak{a}<\mathfrak{u}=\mathfrak{i}$, where the smaller cardinal characteristics have co-analytic witnesses. Our methods allow the preservation of only sufficiently definable witnesses, which significantly differs from other preservation results of this type.