Abstract
Mathematical constructivism could be summarized by the phrase: “Arithmetical statements proven by analytical means can be proven without them, that is by elementary non-analytical means”. Herbrand expressed the idea clearly in the 1920s. It was repeated recently by H. Friedman, following Avigad (2003). The classical example in this connection is Dirichlet who proved in 1837 the prime number theorem and the theorem on arithmetical progressions by analytical means; the elementary proofs were provided only in 1949 by Selberg and Erdös. A less known example is the 1933 Gel’fond-Schneider theorem for which Gel’fond gave a constructive version in the 1960s. I give here a brief account of the theorem and its foundational implications.
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Gauthier, Y. (2015). The Internal Logic of Constructive Mathematics. In: Towards an Arithmetical Logic. Studies in Universal Logic. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-22087-1_6
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