Notre Dame Journal of Formal Logic

Reverse Mathematics and Uniformity in Proofs without Excluded Middle

Jeffry L. Hirst and Carl Mummert

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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

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

First available in Project Euclid: 28 April 2011

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Digital Object Identifier

Zentralblatt MATH identifier

Mathematical Reviews number (MathSciNet)

Primary: 03F35: Second- and higher-order arithmetic and fragments [See also 03B30] 03F50: Metamathematics of constructive systems
Secondary: 03B30: Foundations of classical theories (including reverse mathematics) [See also 03F35] 03F60: Constructive and recursive analysis [See also 03B30, 03D45, 03D78, 26E40, 46S30, 47S30]

reverse mathematics proof theory Dialectica realizability uniformization


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.

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