## On the computational content of the axiom of choice (1998)

Venue: | The Journal of Symbolic Logic |

Citations: | 34 - 1 self |

### BibTeX

@ARTICLE{Berardi98onthe,

author = {Stefano Berardi and Marc Bezem and Thierry Coquand},

title = {On the computational content of the axiom of choice},

journal = {The Journal of Symbolic Logic},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

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.

### Citations

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Citation Context ... of Dependent Choice and Spector's Double Negation Shift. Introduction It is well-known that the Axiom of Choice [15] is validated by theBrouwer{Heyting{Kolmogoro explanation of the logical constants =-=[3]-=-. In view of the negative interpretation of classical arithmetic into intuitionistic arithmetic [6], one would expect that it is possible to make constructive sense of the Axiom of Choice as used in i... |

223 |
Systems that Learn, An Introduction to Learning Theory for Cognitive and Computer Scientists,” MIT-Press
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Citation Context ...mentation", are immediately suggested by trying to understand the computational behaviour of our interpretation. They hint at possible connections with learning theories, such as the one described in =-=[19]-=-, where the learning agent maybene t from negative information. Yet another natural connection is with the work of Hilbert [8, 9] and Ackermann [1]. Our interpretation can be seen as a variation of Hi... |

218 | Mathematical Logic - Shoenfield - 1967 |

122 |
Metamathematical investigations of intuitionistic arithmetic and analysis
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Citation Context ...om schema of induction� (iii) lambda calculus axioms and rules and the de ning equations of the constants R , R xy0 = x, R xy(sz) =yz(R xyz). Thus our theory HA ! essentially coincides with HA ! from =-=[23]-=-, the only di erence being that we consider _ as de ned and use the lambda version instead of the combinator version. The theory HA ! c is HA ! with classical logic� HA ! ;, minimal higher-order arith... |

92 |
What is mathematics
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Citation Context ...luded Middle turns out to be extremely problematic from a constructive point of view. To make constructive sense of such a combination can actually be seen as one the main aims of Hilbert's programme =-=[8, 9]-=-. We address here the more modest question of the analysis of the computational content of the Axiom of Choice, by giving a novel realizability interpretation of the negative translation of the Axiom ... |

82 |
On the interpretation of intuitionistic number theory
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Citation Context ...ds, if we assume GAC� we can prove inHA! ; that not all functions are recursive (even though we cannot exhibit a counterexample!). A direct corollary is that there cannot be any recursive realization =-=[12, 24]-=- of GAC: For exactly this reason, and because the semantics of the system NuPrl is based on recursive realizability, the work [16, 4] restricts itself to a fragment of classical logic that does not in... |

71 |
Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics
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(Show Context)
Citation Context ...tional 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 f... |

62 |
Extracting Constructive Content from Classical Proofs
- Murthy
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(Show Context)
Citation Context ...irect corollary is that there cannot be any recursive realization [12, 24] of GAC: For exactly this reason, and because the semantics of the system NuPrl is based on recursive realizability, the work =-=[16, 4]-=- restricts itself to a fragment of classical logic that does not include the Axiom of Choice. 2 This is to be contrasted with the induction schema over integers, whose negative interpretation is an in... |

44 |
Normal derivability in classical logic
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(Show Context)
Citation Context ...ond-order comprehension, so that one cannot give any predicative interpretation of GAC� a standard computational interpretation of classical arithmetic, the one using in nitary propositional calculus =-=[17, 21]-=-, when extended to quanti cation over functions, fails to interpret the Axiom of Choice. 2.1 G AC refutes Church's Thesis In the classical system HA ! for any statement (x)� we have since, by the Excl... |

43 |
A semantics of evidence for classical arithmetic
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(Show Context)
Citation Context ...ious strategy of 9loise, and the proof that p: p 0 realizes the formula r 9n 8m r f(n) f(m) is similar to the argument showing that this strategy is winning. 7.3 A strategy for the Axiom of Choice In =-=[5]-=- it was conjectured that it should be possible to extend this interpretation in the case of quanti cation over functions. The idea would be simply to allow as index set the set of all functions, and, ... |

29 |
Zermelo’s Axiom of Choice
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(Show Context)
Citation Context ... to compute witnesses from proofs in classical analysis, and how to interpret the axiom of Dependent Choice and Spector's Double Negation Shift. Introduction It is well-known that the Axiom of Choice =-=[15]-=- is validated by theBrouwer{Heyting{Kolmogoro explanation of the logical constants [3]. In view of the negative interpretation of classical arithmetic into intuitionistic arithmetic [6], one would exp... |

25 | Zermelo’s Axiom of Choice. Its Origins, Development and Influence - Moore - 1982 |

22 |
Die logischen Grundlagen der Mathematik. Mathematische Annalen, 88:151–165
- Hilbert
- 1923
(Show Context)
Citation Context ...luded Middle turns out to be extremely problematic from a constructive point of view. To make constructive sense of such a combination can actually be seen as one the main aims of Hilbert's programme =-=[8, 9]-=-. We address here the more modest question of the analysis of the computational content of the Axiom of Choice, by giving a novel realizability interpretation of the negative translation of the Axiom ... |

20 |
Begrtlndung des "Tertium non datur' mittels der Hilbertschen Theorie der Widerspruchsfreiheit. Mathematische Annalen 93
- Ackermann
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(Show Context)
Citation Context ...ith learning theories, such as the one described in [19], where the learning agent maybene t from negative information. Yet another natural connection is with the work of Hilbert [8, 9] and Ackermann =-=[1]-=-. Our interpretation can be seen as a variation of Hilbert's epsilon method [8, 9], with a (classical) proof of termination. It will be interesting to compare our algorithm with the one described in A... |

18 |
Finding computational content in classical proofs
- Constable, Murthy
- 1991
(Show Context)
Citation Context ...irect corollary is that there cannot be any recursive realization [12, 24] of GAC: For exactly this reason, and because the semantics of the system NuPrl is based on recursive realizability, the work =-=[16, 4]-=- restricts itself to a fragment of classical logic that does not include the Axiom of Choice. 2 This is to be contrasted with the induction schema over integers, whose negative interpretation is an in... |

17 | Functional interpretation of bar induction by bar recursion - Howard - 1968 |

14 |
On the principle of excluded middle
- Kolmogorov
- 1977
(Show Context)
Citation Context ...surdity is not interpreted by the empty type, we cannot realize ?) for all . We overcome this problem by exercising some care in the negative interpretation. The idea is essentially due to Kolmogorov =-=[13]-=-. We employ thefact that ?) r can be proved for all without using the axiom schema ?) . Although our prime formulae are decidable, the negative interpretation of a prime formula will be r . As negativ... |

13 |
Normal form theorem for barrecursive functions of type
- Tait
- 1971
(Show Context)
Citation Context ...n that P H [] realizes ?. We give an informal argument, which can easily be formalized using the axiom of Dependent Choice and classical logic. The argument is similar to the argument used by Tait in =-=[22]-=-. Suppose P H [] does not realize ?. Then, by the lemma above, there exist X1�Y1�Z1 such that [(X1�Y1�Z1)] satis es the conditions of the lemma, in particular P H [(X1�Y1�Z1)] does not realize ?. Appl... |

10 | Transfinite induction and bar induction of types zero and one, and the role of continuity in intuitionistic analysis - Howard, Kreisel - 1966 |

5 |
Strong normalization of barrecursive terms without using in terms, Archive for
- Bezem
- 1985
(Show Context)
Citation Context ... interpretation of r � and the inference of r8x : : (x) from8x : :r (x): Closely related should be the question of a constructive formulation of our proof of realizability. Can we adapt the method of =-=[2]-=- and avoid the use of in nite terms? Our hope is that our interpretation, computationally more direct than the one of Spector [18, 10], may provide help for a constructive understanding of classical a... |

1 | Intuitionistic arithmetic as a theory of constructions - Goodman - 1968 |

1 | On the consistency of certain logical calculi - Novikoff - 1943 |

1 |
Trans nite induction and bar induction of types zero and one, and the role of continuity in intuitionistic analysis
- Howard, Kreisel
- 1966
(Show Context)
Citation Context ...intuitionistically its correctness to a principle of bar induction. We also give an interpretation of the Double Negation Shift and try a comparison with Spector's bar recursive interpretation of DNS =-=[18, 10, 11]-=-, which suggests a computational content of the negative interpretation of Axiom of Choice based on Godel's Dialectica translation. We endbyan heuristic explanation of our realizability interpretation... |

1 |
On the consistency of certain logical calculi
- Noviko
- 1943
(Show Context)
Citation Context ...ond-order comprehension, so that one cannot give any predicative interpretation of GAC� a standard computational interpretation of classical arithmetic, the one using in nitary propositional calculus =-=[17, 21]-=-, when extended to quanti cation over functions, fails to interpret the Axiom of Choice. 2.1 G AC refutes Church's Thesis In the classical system HA ! for any statement (x)� we have since, by the Excl... |