Independently Axiomatizable L_{omega_1,omega} Theories

TL;DR

This paper provides conditions under which L_{omega_1,omega} theories can be independently axiomatized, with implications for Vaught's Conjecture and Borel set intersections.

Contribution

It establishes sufficient conditions for independent axiomatization of L_{omega_1,omega} theories and derives related corollaries under set-theoretic assumptions.

Findings

Under Vaught's Conjecture, every countable L_{omega_1,omega} theory has an independent axiomatization.

In ZFC, intersections of families of Borel sets can be represented as intersections of independent Borel sets.

Provides partial answers to questions posed by Miller and Caicedo.

Abstract

In partial answer to a question posed by Arnie Miller (http://www.math.wisc.edu/~miller/res/problem.pdf) and X. Caicedo, we obtain sufficient conditions for an L_{omega_1,omega} theory to have an independent axiomatization. As a consequence we obtain two corollaries: The first, assuming Vaught's Conjecture, every L_{omega_1,omega} theory in a countable language has an independent axiomatization. The second, this time outright in ZFC, every intersection of a family of Borel sets can be formed as the intersection of a family of independent Borel sets.