### Reverse Mathematics and Uniformity in Proofs without Excluded Middle

Jeffry L. Hirst and Carl Mummert
Source: Notre Dame J. Formal Logic Volume 52, Number 2 (2011), 149-162.

#### 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 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.

First Page:
Primary Subjects: 03F35, 03F50
Secondary Subjects: 03B30, 03F60