 
Summary: WEAK THEORIES OF NONSTANDARD
ARITHMETIC AND ANALYSIS
JEREMY AVIGAD
Abstract. A general method of interpreting weak highertype 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 polynomialtime 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 modeltheoretic methods to
build rich extensions of a mathematical structure, like secondorder 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 modeltheoretic arguments, one
can, in general, embed standard mathematical theories is conservative, non
standard extensions. This was done e.g. by Kreisel, for secondorder arithmetic
