Transfinite inductions producing coanalytic sets

TL;DR

This paper generalizes Miller's method of transfinite induction to produce coanalytic sets, demonstrating new constructions such as a coanalytic subset intersecting every $C^1$ curve countably in the constructible universe.

Contribution

It introduces a general condition for modifying transfinite induction to generate coanalytic sets, extending classical results and providing new applications.

Findings

Existence of a coanalytic two-point set, Hamel basis, and MAD family in a consistent framework.

Reproof of classical results using the generalized method.

Construction of an uncountable coanalytic set intersecting every $C^1$ curve countably in $V=L$.

Abstract

A. Miller proved the consistent existence of a coanalytic two-point set, Hamel basis and MAD family. In these cases the classical transfinite induction can be modified to produce a coanalytic set. We generalize his result formulating a condition which can be easily applied in such situations. We reprove the classical results and as a new application we show that in $V=L$ there exists an uncountable coanalytic subset of the plane that intersects every $C^1$ curve in a countable set.