Abstract
Any hereditarily finite set S can be represented as a finite pointed graph –dubbed membership graph– whose nodes denote elements of the transitive closure of {S} and whose edges model the membership relation. Membership graphs must be hyper-extensional, that is pairwise distinct nodes are not bisimilar and (uniquely) represent hereditarily finite sets.We will see that the removal of even a single node or edge from a membership graph can cause “collapses” of different nodes and, therefore, the loss of hyper-extensionality of the graph itself. With the intent of gaining a deeper understanding on the class of hyper-extensional hereditarily finite sets, this paper investigates whether pointed hyper-extensional graphs always contain either a node or an edge whose removal does not disrupt the hyper-extensionality property.
Highlights
A set is hereditarily finite if it is finite and all its elements are hereditarily finite
A hereditarily finite set S can be canonically represented through a pointed finite graph G in which each node represents a different element of the transitive closure of {S} and the edges of G model the membership relation
Since all valid roots of a membership graph belong to the same strongly connected component, our pipeline is able to compute the number of hereditarily finite sets that are represented by graphs of a given order
Summary
A set is hereditarily finite if it is finite and all its elements are hereditarily finite. A hereditarily finite set S can be canonically represented through a pointed finite graph G in which each node represents a different element of the transitive closure of {S} and the edges of G model the membership relation. We provide positive evidence on the fact that there always exists an edge which can be safely removed This result is achieved by introducing the notion of n-well-founded part of a non-well-founded graph and by applying Ackermann code on it.
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