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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.
The Logic of Brouwer and Heyting
, 2007
"... Intuitionistic logic consists of the principles of reasoning which were used informally by ..."
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Intuitionistic logic consists of the principles of reasoning which were used informally by
1 Minima and best approximations in constructive analysis
"... Abstract: Working in Bishop’s constructive mathematics, we first show that minima can be defined as best approximations, in such a way as to preserve the compactness of the underlying metric space when the function is uniformly continuous. Results about finding minima can therefore be carried over ..."
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Abstract: Working in Bishop’s constructive mathematics, we first show that minima can be defined as best approximations, in such a way as to preserve the compactness of the underlying metric space when the function is uniformly continuous. Results about finding minima can therefore be carried over to the setting of finding best approximations. In particular, the implication from having at most one best approximation to having uniformly at most one best approximation is equivalent to Brouwer’s fan theorem for decidable bars. We then show that for the particular case of finitedimensional subspaces of normed spaces, these two notions do coincide. This gives us a better understanding of Bridges ’ proof that finitedimensional subspaces with at most one best approximation do in fact have one. As a complement we briefly review how the case of best approximations to a convex subset of a uniformly convex normed space fits into the unique existence paradigm.