Reverse mathematics and uniformity in proofs without excluded middle

TL;DR

This paper explores how certain constructive analysis statements, provable with intuitionistic logic, have their sequential forms provable in classical systems, using realizability and Dialectica interpretation techniques.

Contribution

It establishes a connection between constructive proofs and classical reverse mathematics through modified realizability and Dialectica interpretation.

Findings

Sequential forms of provable statements are derivable in RCA.

Techniques enable demonstrating unprovability of principles in constructive subsystems.

Results bridge constructive and classical proof frameworks.

Abstract

We show that when certain statements are provable in subsystems of constructive analysis using intuitionistic predicate calculus, related sequential statements are provable in weak classical subsystems. In particular, if a $\Pi^1_2$ sentence of a certain form is provable using E-HA${}^\omega$ along with the axiom of choice and an independence of premise principle, the sequential form of the statement is provable in the classical system RCA. We obtain this and similar results using applications of modified realizability and the \textit{Dialectica} interpretation. These results allow us to use techniques of classical reverse mathematics to demonstrate the unprovability of several mathematical principles in subsystems of constructive analysis.