A Unified Formal Description of Arithmetic and Set Theoretical Data Types

We provide a “shared axiomatization” of natural numbers and hereditarily finite sets built around a polymorphic abstraction of bijective base-2 arithmetics. The “axiomatization” is described as a progressive refinement of Haskell type classes with examples of instances converging to an efficient implementation in terms of arbitrary length integers and bit operations. As an instance, we derive algorithms to perform arithmetic operations efficiently directly with hereditarily finite sets. The self-contained source code of the paper is available at http://logic.cse.unt.edu/tarau/research/2010/unified.hs .