Summary: WEAK THEORIES OF NONSTANDARD
ARITHMETIC AND ANALYSIS
Abstract. A general method of interpreting weak higher-type theories of nonstan-
dard arithmetic in their standard counterparts is presented. In particular, this provides
natural nonstandard conservative extensions of primitive recursive arithmetic, elemen-
tary recursive arithmetic, and polynomial-time computable arithmetic. A means of
formalizing basic real analysis in such theories is sketched.
§1. Introduction. Nonstandard analysis, as developed by Abraham Robin-
son, provides an elegant paradigm for the application of metamathematical
ideas in mathematics. The idea is simple: use model-theoretic methods to
build rich extensions of a mathematical structure, like second-order arithmetic
or a universe of sets; reason about what is true in these enriched structures;
and then transfer the results back to the ordinary mathematical universe.
Robinson showed that this allows one, for example, to provide a coherent and
consistent development of calculus based on the use of infinitesimals.
From a foundational point of view, it is natural to try to axiomatize such
nonstandard structures. By formalizing the model-theoretic arguments, one
can, in general, embed standard mathematical theories is conservative, non-
standard extensions. This was done e.g. by Kreisel, for second-order arithmetic