An interpretation of the Sigma-2 fragment of classical Analysis in System T

Abstract

We show that it is possible to define a realizability interpretation for the $\Sigma_2$-fragment of classical Analysis using G\"odel's System T only. This supplements a previous result of Schwichtenberg regarding bar recursion at types 0 and 1 by showing how to avoid using bar recursion altogether. Our result is proved via a conservative extension of System T with an operator for composable continuations from the theory of programming languages due to Danvy and Filinski. The fragment of Analysis is therefore essentially constructive, even in presence of the full Axiom of Choice schema: Weak Church's Rule holds of it in spite of the fact that it is strong enough to refute the formal arithmetical version of Church's Thesis.