## A. S. Troelstra - Two Papers Related to Constructivism

From: "andrew cooke" <andrew@...>

Date: Thu, 9 Feb 2006 16:44:07 -0300 (CLST)

History of constructivism in the 20th century

Constructivism is a point of view (or an attitude) concerning the methods
and objects of mathematics which is normative: not only does it interpret
existing mathematics according to certain principles, but it also rejects
methods and results not conforming to such principles as unfounded or
speculative (the rejection is not always absolute, but sometimes only a
matter of degree: a decided preference for constructive concepts and
methods)

http://staff.science.uva.nl/~anne/hhhist.pdf

Constructivism and Proof Theory

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.

http://staff.science.uva.nl/~anne/eolss.pdf

First from Derek Elkins on lambda -
http://lambda-the-ultimate.org/node/1264#comment-14372 - although I
believe the article shows that the axiom of infinity was finally rejected
by Bishop (perhaps that is "constructivism" rather than "intuitionism"?).

Andrew