Summary: Interpreting classical theories
in constructive ones
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 second-order arithmetic, bounded arith-
metic, and admissible set theory.
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 GĻodel-Gentzen double-negation 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 Zermelo-Fraenkel set theory in a variant based on intuitionistic logic .
Under the double-negation 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 Friedman-Dragalin interpreta-