Arch. math. Logik 20 (1980), 85-94 CONSERVATIVE EXTENSIONS OF MODELS OF ARITHMETIC* Andreas Blass 1 Abstract We give two characterizations of conservative extensions of models of arith- metic, in terms of the existence and uniqueness of certain amalg~mations with other models. We also establish a connection between conservativity and some combinatorial properties of ultrafilter mappings. Arithmetic is, in this paper, the complete theory of the structure N whose universe is the set of natural numbers and whose relations and functions are all the relations and functions on this set. All models of arithmetic are elementary extensions of N, and, because of the presence of Skolem functions, all submodels of models of arithmetic are elementary submodels. If A _c B are models of arithmetic and if, for every subset X = B that is defnable in B with parameters from B, the intersection Xc~A is definable in A with parameters from A, then B is called a conservative extension of A [9]. For example, because of the definition of N, all its extensions are conservative. In general, all conservative extensions are end extensions [9], but the converse fails, at least if the continuum hypothesis is true [3]. We shall prove two theorems characterizing conservativity in terms of the existence or. uniqueness of certain special amalgamations. We shall also relate conservativity to some combinatorial properties of projections of ultrafilters. Before turning to these results, however, we need to know that models of arithmetic have proper conservative extensions. This can be shown by an iterated ultrapower (or limit ultrapower) argument [7, 33, but the following proof, though perhaps less efficient, seems more conceptual. Let A be any model of arithmetic. A theorem of Keisler [83 says that A is a direct limit of ultrapowers of N with respect to ultrafilters on countable sets. Let *V be the corresponding limit of ultrapowers of the whole set-theoretic universe V. (Readers squeamish about proper classes may truncate V at some reasonably high rank; co 1 is high enough.) For any xE V, we write *x for the corresponding element of * V. Thus, *N is a structure with the same universe as A and with all internal (in * Eingegangen am 14.4.1978. 1 Partially supported by NSF grant MCS 76-06533.