Upwards homogeneity in iterated symmetric extensions

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.