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The paper is organized as a self-contained literate Prolog program thatimplements elements of an executable finite set theory with focus oncombinatorial generation and arithmetic encodings. The complete Prolog code isavailable at http://logic.csci.unt.edu/tarau/research/2008/pHFS.zip . First,ranking and unranking functions for some "mathematically elegant" data types inthe universe of Hereditarily Finite Sets with Urelements are provided,resulting in arithmetic encodings for powersets, hypergraphs, ordinals andchoice functions. After implementing a digraph representation of HereditarilyFinite Sets we define {\em decoration functions} that can recover well-foundedsets from encodings of their associated acyclic digraphs. We conclude with anencoding of arbitrary digraphs and discuss a concept of duality induced by theset membership relation. In the process, we uncover the surprising possibilityof internally sharing isomorphic objects, independently of their language leveltypes and meanings.

Executable Set Theory and Arithmetic Encodings in Prolog