On Petr Novikov's problem of ordered systems of uniform sets

TL;DR

The paper proves that every ordinal less than 2 can be represented as the order type of a system of uniform Borel sets, solving a problem from 1935.

Contribution

It establishes a new connection between ordinal theory and Borel set systems, answering a long-standing open problem.

Findings

Every 2 ordinal is realizable as an order type of a uniform Borel set system.

The result confirms a conjecture posed by Nicolas Luzin in 1935.

Abstract

We prove that every ordinal $\alpha<\omega_2$ is the order type of a certain system of uniform Borel sets in the sense of a well-ordering relation defined by Petr Novikov. This result gives a positive answer to a problem posed by Nicolas Luzin in 1935.