A Dedekind Finite Borel Set

TL;DR

This paper explores properties of Borel sets in models of ZF set theory without the axiom of choice, proving theorems about their structure and constructing models with specific set properties.

Contribution

It establishes new results on Borel sets and constructs models with Dedekind finite sets of reals in ZF, advancing understanding of set theory without choice.

Findings

G-delta-sigma sets are either countable or contain a perfect subset

Existence of uncountable F-sigma-delta sets with no perfect subset under certain conditions

Construction of a model with an infinite Dedekind finite F-sigma-delta set of reals

Abstract

In this paper we prove three theorems about the theory of Borel sets in models of ZF without any form of the axiom of choice. We prove that if B is a G-delta-sigma set, then either B is countable or B contains a perfect subset. Second, we prove that if the real line is the countable union of countable sets, then there exists an F-sigma-delta set which is uncountable but contains no perfect subset. Finally, we construct a model of ZF in which we have an infinite Dedekind finite set of reals which is F-sigma-delta.