arXiv subjects
Publications and source records attributed to Daheng Ju.
Among the four well-known anti-foundation axioms, $\mathsf{BAFA}$, $\mathsf{FAFA}$, $\mathsf{SAFA}$, and $\mathsf{AFA}$, the latter three are all special cases of $\mathsf{AFA}^\sim$ ($\sim$ is one of the regular bisimulations). In this paper, we generalize it to $\mathsf{AFA}^\sim$ ($\sim$ is regular and has the so-called set property), and show that this generalization is substantial by constructing new anti-foundation axioms.
We take the first step toward the study of set-theoretic principles below the axiom of choice $\mathsf{AC}$ and the axiom of foundation $\mathsf{AF}$ by studying Frucht's theorem, an ordinary mathematical theorem which is provable with either $\mathsf{AC}$ or $\mathsf{AF}$ but not provable without both, and its variants. Specifically, we propose a number of such principles, study the relations between these principles and the standard axioms, and prove provability and unprovability results using (infinite) graph-theoretic constructions and permutation models, which draw a preliminary map of this new area of set theory.
In non-well-founded set theory, which anti-foundation axiom is philosophically justified, BAFA, FAFA, SAFA, AFA, or some other? In this paper, we investigate a general approach to answering this question: first, consider which identity condition for sets is justified; second, consider which anti-foundation axiom it justifies. Specifically, we study in detail two plausible identity conditions.