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The paper is organized as a self-contained literate Haskell program thatimplements elements of an executable finite set theory with focus oncombinatorial generation and arithmetic encodings. The code, tested under GHC6.6.1, is available at http://logic.csci.unt.edu/tarau/research/2008/fSET.zip .  We introduce ranking and unranking functions generalizing Ackermann'sencoding to the universe of Hereditarily Finite Sets with Urelements. Then webuild a lazy enumerator for Hereditarily Finite Sets with Urelements thatmatches the unranking function provided by the inverse of Ackermann's encodingand we describe functors between them resulting in arithmetic encodings forpowersets, hypergraphs, ordinals and choice functions. After implementing adigraph representation of Hereditarily Finite Sets we define {\em decorationfunctions} that can recover well-founded sets from encodings of theirassociated acyclic digraphs. We conclude with an encoding of arbitrary digraphsand discuss a concept of duality induced by the set membership relation.  Keywords: hereditarily finite sets, ranking and unranking functions,executable set theory, arithmetic encodings, Haskell data representations,functional programming and computational mathematics

A Functional Hitchhiker's Guide to Hereditarily Finite Sets, Ackermann Encodings and Pairing Functions