In this note we show that the so-called weakly extensional arithmetic in all nite types, which is based on a quanti er-free rule of extensionality due to C. Spector and which is of signi cance in the context of Godel's functional interpretation, does not satisfy the deduction theorem for additional axioms. This holds already for 1 - axioms. Previously, only the failure of the stronger deduction theorem for deductions from (possibly open) assumptions (with parameters kept xed) was known.

In [12], the second author obtained metatheorems for the extraction of effective (uniform) bounds from classical, prima facie nonconstructive proofs in functional analysis. These metatheorems for the first time cover general classes of structures like arbitrary metric, hyperbolic, CAT(0) and normed linear spaces and guarantee the independence of the bounds from parameters raging over metrically bounded (not necessarily compact!) spaces. The use of classical logic imposes some severe restrictions on the formulas and proofs for which the extraction can be carried out. In this paper we consider similar metatheorems for semi-intuitionistic proofs, i.e. proofs in an intuitionistic setting enriched with certain non-constructive principles. Contrary to

We show that the so-called weak Markov's principle (WMP) which states that every pseudo-positive real number is positive is underivable in T # :=EHA # +AC. Since T # allows to formalize (at least large parts of) Bishop's constructive mathematics this makes it unlikely that WMP can be proved within the framework of Bishop-style mathematics (which has been open for about 20 years). The underivability even holds if the ine#ective schema of full comprehension (in all types) for negated formulas (in particular for #-free formulas) is added which allows to derive the law of excluded middle for such formulas.

Abstract. We show how different functional interpretations can be combined via a multi-modal linear logic. A concrete hybrid of Kreisel’s modified realizability and Gödel’s Dialectica is presented, and several small applications are given. We also discuss how the hybrid interpretation relates to variants of Dialectica and modified realizability with non-computational quantifiers. 1

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Abstract. The purpose of this article is to present a parametrised functional interpretation. Depending on the choice of the parameter relations one obtains well-known functional interpretations, such as Gödel’s Dialectica interpretation, Diller-Nahm’s variant of the Dialectica interpretation, Kohlenbach’s monotone interpretations, Kreisel’s modified realizability and Stein’s family of functional interpretations. A functional interpretation consists of a formula translation and a proof translation. We show that all these interpretation only differ on two choices: firstly, on “how much ” of the counterexamples for A became witnesses for ¬A when defining the formula translation, and, secondly, “how much ” of the witnesses of A one is interested in when defining the proof translation.

an optimization of Gödel’s technique towards the extraction of (more) efficient programs from (classical) proofs

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this article has appeared in Computational Logic and Proof Theory (Proc. 3

Abstract not found