 
Summary: Interpreting classical theories
in constructive ones
Jeremy Avigad
Abstract
A number of classical theories are interpreted in analogous theories
that are based on intuitionistic logic. The classical theories considered
include subsystems of first and secondorder arithmetic, bounded arith
metic, and admissible set theory.
1 Introduction
Proof theory was developed, in part, as a way to reconcile classical and con
structive aspects of mathematical reasoning. Given this historical fact, it is not
surprising that over the years proof theorists have invested a good deal of effort
in reducing classical theories to constructive ones.
Elegant in its simplicity, the GĻodelGentzen doublenegation interpretation
works wonders. For example, it seamlessly reduces classical arithmetic to its
intuitionistic counterpart, and with some additional work it can be used to in
terpret ZermeloFraenkel set theory in a variant based on intuitionistic logic [13].
Under the doublenegation interpretation, 2 sentences of arithmetic, which can
be said to carry a theory's "computational" content, are not preserved, but in
both these examples a further application of the FriedmanDragalin interpreta
