Abstract. We survey some research aiming at a theory of effective structures of size the continuum. The main notion is the one of a Borel presentation, where the domain, equality and further relations and functions are Borel. We include the case of uncountable languages where the signature is Borel. We discuss the main open questions in the area. 1.

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.

This paper develops the model theory of ordered structures that satisfy the analogue, REF (L), of the reflection principle of Zermelo-Fraenkel set theory. Here L is a language with a distinguished linear order <, and REF (L) consists of the universal closure of formulas of the form ∃x∀y1 < x · · · ∀y1 < x ϕ(y1, · · ·, yn) ↔ ϕ <x (y1, · · ·, yn), where ϕ(y1, · · ·, yn) is an L-formula, ϕ <x is the L-formula obtained by restricting all the quantifiers of ϕ to the initial segment determined by x, and x is a variable that does appear in ϕ. Our results include: Theorem. The following five conditions are equivalent for a complete first order theory T in a countable language L with a distinguished linear order: (1) Some model of T has an elementary end extension with a first new element. (2) T ⊢ REF (L). (3) T has an ω1-like model that continuously embeds ω1. (4) For some regular uncountable cardinal κ, T has a κ-like model that continuously embeds a stationary subset of κ. (5) For some regular uncountable cardinal κ, T has a κ-like model M that has an elementary extension in which the supremum of M exists. Moreover, if κ is a regular cardinal satisfying κ = κ <κ, then each of the above conditions is equivalent to: (6) T has a κ +-like model that continuously embeds a stationary subset of κ.

1 Descriptive use of logic and Intended models 1 1.1 Standard models of arithmetic.......................... 1 1.2 Axiomatics and Formal theories......................... 3 1.3 Hintikka and the two uses of logic in mathematics.............. 5

We prove that N has an uncountable elementary extension N such that there is no ultrafilter on the Boolean Algebra of subsets of N represented in N which is minimal (i.e. Ramsey for partitions represented in N).