Notre Dame Journal of Formal Logic

Reverse Mathematics and Uniformity in Proofs without Excluded Middle

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.

Article information

Source
Notre Dame J. Formal Logic Volume 52, Number 2 (2011), 149-162.

Dates
First available in Project Euclid: 28 April 2011

http://projecteuclid.org/euclid.ndjfl/1303995711

Digital Object Identifier
doi:10.1215/00294527-1306163

Mathematical Reviews number (MathSciNet)
MR2794648

Zentralblatt MATH identifier
05903675

Citation

Hirst, Jeffry L.; Mummert, Carl. Reverse Mathematics and Uniformity in Proofs without Excluded Middle. Notre Dame J. Formal Logic 52 (2011), no. 2, 149--162. doi:10.1215/00294527-1306163. http://projecteuclid.org/euclid.ndjfl/1303995711.