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Constructivism and Proof Theory
, 2003
"... 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. F ..."
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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 socalled Gentzen systems which use
sequents instead of formulas and his result on firstorder arithmetic showing that
(suitably formalized) transfinite induction up to the ordinal "0 cannot be proved in
firstorder arithmetic.
Marginalia on Sequent Calculi
, 1998
"... Introduction In this note we wish to draw attention to certain points of detail, concerning the subtle differences between several possible versions of Gentzen systems and systems of natural deduction. In notation and terminology we conform to [TS96]. Gentzen([Gen35]) introduced the calculi LJ, LK ..."
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Introduction In this note we wish to draw attention to certain points of detail, concerning the subtle differences between several possible versions of Gentzen systems and systems of natural deduction. In notation and terminology we conform to [TS96]. Gentzen([Gen35]) introduced the calculi LJ, LK with left and right introduction rules, operating on sequents; systems of this kind are in the literature often called "sequent calculi". Contrary to what many people think, natural deduction did not originate with Gentzen (there is, for example, the earlier work by J'askowski, [J'as34]) although Gentzen's work made it wellknown. Versions of natural deduction are sometimes also presented as calculi operating on sequents (as in this note). For these reasons I have rejected the widely used designation "sequent calculi" for systems of the LJ,LKtype and call them "Gentzen systems" instead. (One might object that "Gentzen systems" might be interpreted as referring to all the formalisms