Cardinal Well-foundedness and Choice
Andreas Blass, Dhruv Kulshreshtha
TL;DR
This work analyzes several notions of cardinal well-foundedness in ZF without the Axiom of Choice, clarifying distinctions and interdependencies among them. It develops a formal framework for WF notions $\mathsf{WF}_{yk}^x$, investigates their implications, and connects them to major choice principles such as CC, DC, CSB^*, and PP; notably, surjective well-foundedness implies the Dual Cantor-Schröder-Bernstein theorem. It shows that the Partition Principle would collapse all WF notions to equivalence, while the question of whether any WF form implies AC remains open, underscoring the delicate relationship between cardinal structure and choice. By combining results on Dedekind sets, cardinal representatives, and countable-choice variants, the paper maps a rich landscape of logical interactions in the absence of AC and offers concrete open problems to guide future research.
Abstract
We consider several notions of well-foundedness of cardinals in the absence of the Axiom of Choice. Some of these have been conflated by some authors, but we separate them carefully. We then consider implications among these, and also between these and other consequences of Choice. For instance, we show that the Partition Principle implies that all of our versions of well-foundedness are equivalent. We also show that one version, concerning surjections, implies the Dual Cantor-Schröder-Bernstein theorem. It has been conjectured that well-foundedness, in one form or another, actually implies the Axiom of Choice, but this conjecture remains unresolved.