Iterating Symmetric Extensions

TL;DR

This paper develops a framework for iterating symmetric extensions to construct new models of ZF, enabling the analysis of choice principles and their failures in set theory.

Contribution

It introduces a novel framework for iterating symmetric extensions, expanding the tools for constructing and analyzing models of ZF with specific choice properties.

Findings

Framework for iterating symmetric extensions developed

Results on Kinna--Wagner principles and their failure obtained

New models of ZF constructed using the framework

Abstract

The notion of a symmetric extension extends the usual notion of forcing by identifying a particular class of names which forms an intermediate model of ZF between the ground model and the generic extension, and often the axiom of choice fails in these models. Symmetric extensions are generally used to prove choiceless consistency results. We develop a framework for iterating symmetric extensions in order to construct new models of ZF. We show how to obtain some well-known and lesser-known results using this framework. Specifically, we discuss Kinna--Wagner principles and obtain some results related to their failure.