The Class of Countable Projective Planes is Borel Complete

TL;DR

This paper proves that the class of countable projective planes, including non-Desarguesian and Pappian ones, is Borel complete, highlighting the complexity of their classification within descriptive set theory.

Contribution

It demonstrates the Borel completeness of the class of countable projective planes, extending previous results and applying classical projective geometry techniques.

Findings

The class of countable non-Desarguesian projective planes is Borel complete.

The class of countable Pappian projective planes is Borel complete.

Every group can be realized as the collineation group of some projective plane.

Abstract

We observe that Hall's free projective extension $P \mapsto F(P)$ of partial planes is a Borel map, and use a modification of the construction introduced in [9] to conclude that the class of countable non-Desarguesian projective planes is Borel complete. In the process, we also rediscover the main result of [7] on the realizability of every group as the group of collineations of some projective plane. Finally, we use classical results of projective geometry to prove that the class of countable Pappian projective planes is Borel complete.