On double-membership graphs of models of Anti-Foundation

TL;DR

This work studies double-membership graphs derived from countable models of Anti-Foundation by examining the D-graph $M_1$ and the SD-graph $M_0$ associated to a model $M\vDash \mathsf{ZFA}$. It shows that connected components of $M_1$ align with unions of regions in $M$ and, in contrast, that without Foundation these graphs can realize essentially any graph, yielding a wealth of non-well-founded structures. The authors establish continuum-many non-isomorphic D-graphs and continuum-many countable models of each theory, using $n$-flowers and $A$-bouquets to encode local configurations and flat systems under $\mathsf{AFA}$ to realize them, and they prove that the common theory $\mathrm{Th}(K_1)$ is incomplete with completions characterized by consistent collections of consistency statements, all of which are model-theoretically wild. They further derive consequences such as the existence of countable models elementarily equivalent to a D-graph but not arising from any $\mathsf{ZFA}$-model, and a no-infinite-diameter result that yields SD-graphs not obtainable as D-graphs, providing negative answers to questions in earlier work on these graphs. Overall, the paper develops a robust model-theoretic picture of non-well-founded graph reducts, employing EF games, interpretability arguments, and finite-structure reasoning to map the landscape of completions and their complexity.

Abstract

We answer some questions about graphs which are reducts of countable models of Anti-Foundation, obtained by considering the binary relation of double-membership $x\in y\in x$. We show that there are continuum-many such graphs, and study their connected components. We describe their complete theories and prove that each has continuum-many countable models, some of which are not reducts of models of Anti-Foundation.

Figures (2)

• Figure 1: On the left, a picture of the unique sets $a$ and $b$ such that $a=\{b, \emptyset\}$ and $b=\{a,\{\emptyset\}\}$. On the right, a picture of the unique set $c$ such that $c=\{c,\emptyset, \{\emptyset\}\}$. The arrows denote membership.
• Figure 2: The set $a=\{\{a,i\}\mid i<5\}$ is a $5$-flower. The reason for the name '$n$-flower' can be seen in this figure.

Theorems & Definitions (49)

• Theorem : Corollary \ref{['co:manymodels']}
• Theorem : Theorem \ref{['thm:completions']}
• Theorem : Corollary \ref{['co:q5']}
• Definition 1.1
• Example 1.2
• Remark 1.3
• proof
• Definition 1.6
• Definition 1.7
• Definition 2.1