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On the computational content of the axiom of choice
 The Journal of Symbolic Logic
, 1998
"... 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 rea ..."
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Cited by 34 (1 self)
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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.
On the NoCounterexample Interpretation
 J. SYMBOLIC LOGIC
, 1997
"... In [15],[16] Kreisel introduced the nocounterexample 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 firstorder Peano arithmetic PA one can find ordinal recursive functi ..."
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Cited by 18 (10 self)
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In [15],[16] Kreisel introduced the nocounterexample 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 firstorder 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
Representations of stream processors using nested fixed points
 Logical Methods in Computer Science
"... Abstract. We define representations of continuous functions on infinite streams of discrete values, both in the case of discretevalued functions, and in the case of streamvalued functions. We define also an operation on the representations of two continuous functions between streams that yields a ..."
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Cited by 15 (2 self)
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Abstract. We define representations of continuous functions on infinite streams of discrete values, both in the case of discretevalued functions, and in the case of streamvalued 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 discretevalued functions, the representatives are wellfounded (finitepath) trees of a certain kind. The underlying idea can be traced back to Brouwer’s justification of barinduction, or to Kreisel and Troelstra’s elimination of choicesequences. In the case of streamvalued functions, the representatives are nonwellfounded trees pieced together in a coinductive fashion from wellfounded trees. The definition requires an alternating fixpoint construction of some ubiquity.
The Borel hierarchy and the projective hierarchy in intuitionistic mathematics
 University of Nijmegen Department of Mathematics
, 2001
"... 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 ma ..."
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Cited by 4 (2 self)
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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
Almost The Fan Theorem
, 2001
"... This paper is a slightly revised translation of [20] We calculate n such that, for every y in [0, 1], if 2 m+1 . Finally we find i such that i > n and 2 n+1 . We conclude: xy 2 n , and therefore: i )f(y i ) < f(x)) f(y i )f(x) < 2 m . Contradiction. T ..."
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Cited by 1 (0 self)
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This paper is a slightly revised translation of [20] We calculate n such that, for every y in [0, 1], if 2 m+1 . Finally we find i such that i > n and 2 n+1 . We conclude: xy 2 n , and therefore: i )f(y i ) < f(x)) f(y i )f(x) < 2 m . Contradiction. There must exist a suitable n
REPRESENTATIONS OF STREAM PROCESSORS USING NESTED FIXED POINTS
, 2008
"... Vol. 5 (3:9) 2009, pp. 1–17 www.lmcsonline.org ..."