Borel two-point sets — Problem 5.1

5. Jialiang He

These notes concern two related definability questions in infinite combinatorics and Borel model theory.

Definition 5.1.

A set $X\subseteq\mathbb{R}^{2}$ is called a two-point set, if for every line $L\subseteq\mathbb{R}^{2}$,

Theorem 5.2 (Larman [1]).

There is no $F_{\sigma}$ two-point set.

Since every $F_{\sigma}$ set is Borel, Larman’s theorem rules out the lowest nontrivial possibility in the Borel hierarchy, but it does not settle the following problem.

Miller showed that, assuming Gödel’s axiom of constructibility $V=L$, there is a $\boldsymbol{\Pi}^{1}_{1}$, i.e., coanalytic two-point set [2].

Definition 5.3.

A standard Borel space is a measurable space isomorphic to the Borel space of a Polish space. A Borel field is a field $(F,+,\times)$ whose universe is a standard Borel space and whose field operations $+,\times$ are Borel.

Definition 5.4.

Let $F$ be a Borel field. A Borel algebraically closed extension of $F$ is a Borel field $K$, together with a Borel injective field homomorphism

such that $K$ is algebraically closed.

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