Dynamical ideals and the axiom of choice
Jindrich Zapletal
TL;DR
The paper studies how natural group-action properties on atom structures yield fragments of the axiom of choice in choiceless permutation models. It introduces dynamical ideals $(\Gamma\curvearrowright X,I)$ and their associated models $W[[X]]$, and develops dynamical criteria—cofinal orbits, DC-completeness, dynamic $\kappa$-completeness, and simplicity—that correspond to WO-C, DC, $\mathrm{AC}_\kappa$, and related choice principles. It provides a suite of equivalence results and a breadth of examples from topology, model theory, and Fraïssé limits, demonstrating how symmetry constrains choice and shapes the structure of admissible sets. The findings illuminate the interplay between dynamics and choice in choiceless set theory, with implications for the existence of amorphous or Dedekind-finite sets and for understanding how stratification and simplicity govern the landscape of permutation-model constructions.
Abstract
I provide several natural properties of group actions which translate into fragments of axiom of choice in the associated permutation models of choiceless set theory.