Upwards homogeneity in iterated symmetric extensions

TL;DR

This paper investigates conditions under which iterated symmetric extensions in set theory do not add new sets of ordinals beyond the first extension, focusing on a property called upwards homogeneity.

Contribution

It identifies the precise conditions for upwards homogeneity in iterated symmetric extensions and demonstrates its applicability to various known constructions.

Findings

Characterization of conditions for upwards homogeneity

Application to multiple known set-theoretic constructions

Framework for constructing models with controlled set addition

Abstract

It is sometimes desirable in choiceless constructions of set theory that one iteratively extends some ground model without adding new sets of ordinals after the first extension. Pushing this further, one may wish to have models $V \subseteq M \subseteq N$ of $\mathsf{ZF}$ such that $N$ contains no subsets of $V$ that do not already appear in $M$. We isolate, in the case that $M$ and $N$ are symmetric extensions (particular inner models of a generic extension of $V$), the exact conditions that cause this behaviour and show how it can broadly be applied to many known constructions. We call this behaviour upwards homogeneity.