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A Realizability Interpretation for Classical Arithmetic

Summary: A Realizability Interpretation for
Classical Arithmetic
Jeremy Avigad
Carnegie Mellon University, Pittsburgh, PA 15231, USA
Summary. A constructive realizablity interpretation for classical arithmetic is presented,
enabling one to extract witnessing terms from proofs of 1 sentences. The interpretation
is shown to coincide with modified realizability, under a novel translation of classical logic
to intuitionistic logic, followed by the Friedman-Dragalin translation. On the other hand,
a natural set of reductions for classical arithmetic is shown to be compatible with the
normalization of the realizing term, implying that certain strategies for eliminating cuts
and extracting a witness from the proof of a 1 sentence are insensitive to the order in
which reductions are applied.
1 Introduction
Even though, as is well known, the classical and intuitionistic versions of first-
order arithmetic prove the same 2 sentences, the two theories are very different
in nature. In particular, the intuitionistic version has a constructive interpretation
which seems to be lacking in its classical counterpart.
Heyting arithmetic, which is based on intuitionistic logic, is perhaps best
represented in a system of natural deduction. In this framework, proofs can be
associated with (or construed as) realizing terms, which come equipped with a


Source: Avigad, Jeremy - Departments of Mathematical Sciences & Philosophy, Carnegie Mellon University


Collections: Multidisciplinary Databases and Resources; Mathematics