Abstract. Finite set theory, here denoted ZFfin, is the theory obtained by replacing the axiom of infinity by its negation in the usual axiomatization of ZF (Zermelo-Fraenkel set theory). An ω-model of ZFfin is a model in which every set has at most finitely many elements (as viewed externally). Mancini and Zambella (2001) employed the Bernays-Rieger method of permutations to construct a recursive ω-model of ZFfin that is nonstandard (i.e., not isomorphic to the hereditarily finite sets Vω). In this paper we initiate the metamathematical investigation of ω-models of ZFfin. In particular, we present a perspicuous method for constructing recursive nonstandard ω-model of ZFfin without the use of permutations. We then use this method to establish the following central theorem. Theorem A. For every simple graph (A, F), where F is a set of unordered pairs of A, there is an ω-model M of ZFfin whose universe contains A and which satisfies the following two conditions: (1) There is parameter-free formula ϕ(x, y) such that for all elements a and b of M, M | = ϕ(a, b) iff {a, b} ∈ F; (2) Every element of M is definable in (M, c)c∈A. Theorem A enables us to build a variety of ω-models with special features, in particular: Corollary 1. Every group can be realized as the automorphism group of an ω-model of ZFfin. Corollary 2. For each infinite cardinal κ there are 2 κ rigid nonisomorphic ω-models of ZFfin of cardinality κ. Corollary 3. There are continuum-many nonisomorphic pointwise definable ω-models of ZFfin.

Abstract. We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higher-order reflection principles. Paul Isaak Bernays (1888–1977) is an important figure in the development of mathematical logic, being the main bridge between Hilbert and Gödel in the intermediate generation and making contributions in proof theory, set theory, and the philosophy of mathematics. Bernays is best known for the two-volume 1934,1939 Grundlagen der Mathematik [39, 40], written solely by him though Hilbert was retained as first author. Going into many reprintings and an eventual second edition thirty years later, this monumental work provided a magisterial exposition of the work of the Hilbert school in the formalization of first-order logic and in proof theory and the work of Gödel on incompleteness and its surround, including the first complete proof of the Second Incompleteness Theorem. 1 Recent re-evaluation of Bernays ’ role actually places him at the center of the development of mathematical logic and Hilbert’s program. 2 But starting in his forties, Bernays did his most individuated, distinctive mathematical work in set theory, providing a timely axiomatization and later applying higher-order reflection principles, and produced a stream of