The hierarchy of omega1-Borel sets
Arnold W. Miller
TL;DR
This paper investigates the structure of omega1-Borel sets, establishing the length of their hierarchy under different set-theoretic assumptions, revealing its dependence on models of set theory.
Contribution
It proves the length of the omega1-Borel hierarchy is omega2 under MA+notCH and determines its possible lengths in the Cohen real model.
Findings
Hierarchy length is omega2 under MA+notCH.
In Cohen real model, hierarchy length is either omega1+1 or omega1+2.
Abstract
The family omega1-Borel sets is the smallest family of subsets of the real line which contains the family of open sets and is closed under complementation and omega1 unions. We show: Theorem 1. MA+notCH implies this hierarchy has length omega2. Theorem 2. In the Cohen real model it has length either omega1+1 or omega1+2.
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Taxonomy
TopicsAdvanced Topology and Set Theory · Fuzzy and Soft Set Theory · Rings, Modules, and Algebras