Mapping Sets and Hypersets into Numbers

We introduce and prove the basic properties of encodings that generalize to non-well-founded hereditarily finite sets the bijection defined by Ackermann in 1937 between hereditarily finite sets and natural numbers

Hereditarily finite sets and identity trees

AbstractSome asymptotic results about the sizes of certain sets of hereditarily finite sets, identity trees, and finite games are proven

Is Hyper-extensionality Preservable Under Deletions of Graph Elements?

Any hereditarily finite set S can be represented as a finite pointed graph \u2013dubbed membership graph\u2013 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 \u201ccollapses\u201d 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

Safe Recursive Set Functions

This paper introduces the safe recursive set functions based on a Bellantoni-Cook style subclass of the primitive recursive set functions. It shows that the functions computed by safe recursive set functions under a list encoding of finite strings by hereditarily finite sets are exactly the polynomial growth rate functions computed by alternating exponential time Turing machines with polynomially many alternations. It also shows that the functions computed by safe recursive set functions under a more efficient binary tree encoding of finite strings by hereditarily finite sets are exactly the quasipolynomial growth rate functions computed by alternating quasipolynomial time Turing machines with polylogarithmic many alternations. The safe recursive set functions are characterized on arbitrary sets in definability-theoretic terms. In its strongest form, it is shown that a function on arbitrary sets is safe recursive if, and only if, it is uniformly definable in some polynomial level of a refinement of Jensen's J-hierarchy, relativised to the transitive closure of the function's arguments. An observation is that safe-recursive functions on infinite binary strings are equivalent to functions computed by so-called infinite-time Turing machines in time less than ωω. Finally a machine model is given for safe recursion which is based on set-indexed parallel processors and the natural bound on running times