I argue for the use of the adjunction operator (adding a single new element to an existing set) as a basis for building a finitary set theory. It allows a simplified axiomatization for the first-order theory of hereditarily finite sets based on an induction schema and a rigorous characterization of the primitive recursive set functions. The latter leads to a primitive recursive presentation of arithmetical operations on finite sets.

Ordinal addition and multiplication can be extended in a natural way to all sets. I survey the structure of the sets under these operations. In particular, the natural partial ordering associated with addition of sets is shown to be a tree. This allows us to prove that any set has a unique representation as a sum of additively irreducible sets, and that the non-empty elements of any model of set theory can be partitioned into infinitely many submodels, each isomorphic to (...) the original model. Also any model of set theory has an isomorphic extension in which the empty set of the original model is non-empty. Among other results, the relations between the arithmetical operations and the transitive closure and the adductive hierarchy are elucidated. (shrink)

This article defines a hierarchy on the hereditarily finite sets which reflects the way sets are built up from the empty set by repeated adjunction, the addition to an already existing set of a single new element drawn from the already existing sets. The structure of the lowest levels of this hierarchy is examined, and some results are obtained about the cardinalities of levels of the hierarchy.

Any set x is uniquely specified by the graph of the membership relation on the set obtained by adjoining x to the transitive closure of x. Thus any operation on sets can be looked at as an operation on these graphs. We look at the operations of ordinal arithmetic of sets in this light. This turns out to be simplest for a modified ordinal arithmetic based on the Zermelo ordinals, instead of the usual von Neumann ordinals. In this arithmetic, addition (...) of sets corresponds to concatenating graphs, and multiplication corresponds to replacing each edge of a graph by a copy of another graph. Characterizations for the von Neumann case are also given. (shrink)

The “high school algebra” laws of exponentiation fail in the ordinal arithmetic of sets that generalizes the arithmetic of the von Neumann ordinals. The situation can be remedied by using an alternative arithmetic of sets, based on the Zermelo ordinals, where the high school laws hold. In fact the Zermelo arithmetic of sets is uniquely characterized by its satisfying the high school laws together with basic properties of addition and multiplication. We also show how in both arithmetics the behavior of (...) exponentiation depends on whether the empty set is an element of the base. (shrink)

A survey of the isomorphic submodels of Vω, the set of hereditarily finite sets. In the usual language of set theory, Vω has 2ℵ0 isomorphic submodels. But other set-theoretic languages give different systems of submodels. For example, the language of adjunction allows only countably many isomorphic submodels of Vω.

We define an axiom schema for finite set theory with bounded induction on sets, analogous to the theory of bounded arithmetic,, and use some of its basic model theory to establish some independence results for various axioms of set theory over. Then we ask: given a model M of, is there a model of whose ordinal arithmetic is isomorphic to M? We show that the answer is yes if.