Limit-stage behavior of finite-support iterations of upwards homogeneous symmetric extensions

Ascertain the structural and preservation properties of the limit-stage model obtained from a finite-support iteration of upwards homogeneous symmetric extensions, particularly beyond ω-length, and determine what can be said about the resulting model at the limit stage when each iterand is upwards homogeneous over its predecessor.

Background

The paper exhibits ω-length iterations where each step is upwards homogeneous over its predecessor, ensuring desirable preservation properties across finite partial factorizations. Extending this behavior to longer iterations and understanding the model at limit stages requires new techniques.

Recent work (Shani, 2021) provides tools to move beyond ω steps in iterated symmetric extensions. The authors ask specifically what can be concluded about the limit-stage model of a finite-support iteration under upwards homogeneity, aiming to generalize and clarify the behavior of such iterations at limits.