Amphi-ZF : axioms for Conway games

A theory of two-sided containers, denoted ZF₂, is introduced. This theory is then shown to be synonymous to ZF in the sense of Visser (2006), via an interpretation involving Quine pairs. Several subtheories of ZF₂, and their relationships with ZF, are also examined. We include a short discussion of permutation models (in the sense of Rieger–Bernays) over ZF₂. We close with highlighting some areas for future research, mostly motivated by the need to understand non-wellfounded games.