We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice. Our interpretation seems computationally more direct than the one based on Godel's Dialectica interpretation [10, 18]. Interestingly, thisinterpretation uses a re nement of the realizibility semantics of the absurdity proposition, which is not interpreted as the empty type here. We alsoshowhow to compute witnesses from proofs in classical analysis, and how to interpret the axiom of Dependent Choice and Spector's Double Negation Shift.

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In [15],[16] Kreisel introduced the no-counterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated "-substitution method (due to W. Ackermann), that for every theorem A (A prenex) of first-order Peano arithmetic PA one can find ordinal recursive functionals \Phi A of order type ! " 0 which realize the Herbrand normal form A of A. Subsequently more

Abstract. We define representations of continuous functions on infinite streams of discrete values, both in the case of discrete-valued functions, and in the case of stream-valued functions. We define also an operation on the representations of two continuous functions between streams that yields a representation of their composite. In the case of discrete-valued functions, the representatives are well-founded (finitepath) trees of a certain kind. The underlying idea can be traced back to Brouwer’s justification of bar-induction, or to Kreisel and Troelstra’s elimination of choice-sequences. In the case of stream-valued functions, the representatives are non-wellfounded trees pieced together in a coinductive fashion from well-founded trees. The definition requires an alternating fixpoint construction of some ubiquity.

this paper. Once we agree to accept and use it we enter a new world and discover many facts for which there does not exist a classical counterpart. The principle entails for instance that the union of the two closed sets [0, 1] and [1, 2] is not a countable intersection of open subsets of R. One may also infer that there are unions of three closed sets di#erent from every union of two closed sets. These observations are the tip of an iceberg. The intuitionistic Borel Hierarchy shows o# an exquisite fine structure

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Vol. 5 (3:9) 2009, pp. 1–17 www.lmcs-online.org