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Constructivism and Proof Theory (2003)
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BibTeX
@MISC{Troelstra03constructivismand,
author = {A. S. Troelstra},
title = {Constructivism and Proof Theory},
year = {2003}
}
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Abstract
Introduction to the constructive point of view in the foundations of mathematics, in particular intuitionism due to L.E.J. Brouwer, constructive recursive mathematics due to A.A. Markov, and Bishop’s constructive mathematics. The constructive interpretation and formalization of logic is described. For constructive (intuitionistic) arithmetic, Kleene’s realizability interpretation is given; this provides an example of the possibility of a constructive mathematical practice which diverges from classical mathematics. The crucial notion in intuitionistic analysis, choice sequence, is briefly described and some principles which are valid for choice sequences are discussed. The second half of the article deals with some aspects of proof theory, i.e., the study of formal proofs as combinatorial objects. Gentzen’s fundamental contributions are outlined: his introduction of the so-called Gentzen systems which use sequents instead of formulas and his result on first-order arithmetic showing that (suitably formalized) transfinite induction up to the ordinal "0 cannot be proved in first-order arithmetic.
Keyphrases
proof theory choice sequence constructive recursive mathematics constructive mathematics realizability interpretation intuitionistic analysis transfinite induction constructive mathematical practice crucial notion constructive interpretation so-called gentzen system particular intuitionism a.a. markov classical mathematics briefly described fundamental contribution formal proof article deal constructive point second half l.e.j. brouwer combinatorial object first-order arithmetic showing
