Abstract

This paper essentially contains material from chapter 8 of the author's Habilitationschrift. Some of the results were presented at the Logic Colloquium 94 at Clermont--Ferrand (see [7]). 1 As is well-known (cf. the discussion at the end of x3 of [10]), the use of classical logic (on which the systems GnA ! are based) has the consequence that the extractability of an effective (and for n = 2 polynomial) bound from a proof of an 89A--sentence is (in general) guaranteed only if A is quantifier-- free (or purely existential). In the present paper we study proofs which may use mathematically strong non--constructive analytical principles as e.g. 1) Attainment of the maximum of f 2 C([0; 1] d ; IR) 2) Mean value theorem for integrals 3) Cauchy--Peano existence theorem 4) Brouwer's fixed point theorem for continuous functions f : [0; 1] d ! [0; 1] d 5) A generalization WKL 2 seq of the binary Konig's lemma WKL 6) Comprehension for negated formulas: CA ae : : 9\Phi 0ae x ae :1 0 8y ae \Gamma \Phiy = 0 0 $ :A(y) \Delta ; where A is arbitrary: as well as the non-intuitionistic logical principles 7) The `double negation shift' DNS : 8x ae ::A ! ::8x ae A for arbitrary types ae and formulas A 8) The `lesser limited principle of omniscience' LLPO : 8x 1 ; y 1 9k 0 1([k = 0 ! x IR y] [k = 1 ! y IR x]) 9) The independence of premise principle for negated formulas IP: : (:A ! 9y ae B) ! 9y ae (:A ! B); where y is not free in A, plus the schema AC of full choice but apply these principles only in the context of the intuitionistic versions (E)--GnA ! i of the theories (E)--G n A ! . The restriction to intuitionistic logic guarantees the extractability of (uniform) effective bounds for arbitrary 89A--sentences (see theorem 4.1 below). Indeed w...

Citations

45	Varieties of constructive mathematics - Bridges, Richman - 1987
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2	Principles of continuous choice and continuity of functions in formal systems for constructive mathematics - Beeson - 1977