## Bounded Modified Realizability (2005)

Citations: | 1 - 1 self |

### BibTeX

@MISC{Ferreira05boundedmodified,

author = {Fernando Ferreira and Ana Nunes},

title = {Bounded Modified Realizability},

year = {2005}

}

### OpenURL

### Abstract

We define a notion of realizability, based on a new assignment of formulas, which does not care for precise witnesses of existential statements, but only for bounds for them. The novel form of realizability supports a very general form of the FAN theorem, refutes Markov’s principle but meshes well with some classical principles, including the lesser limited principle of omniscience and weak König’s lemma. We discuss some applications, as well as some previous results in the literature. 1

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Citation Context ...uitionistic logic (see Subsections 4.2 and 4.3). In order to deal with full classical reasoning, one must use the more sophisticated tool of the functional interpretation, introduced by Kurt Gödel in =-=[6]-=- (one should add, for quite different reasons – see the discussion in [14]). Contrary to realizability, Gödel’s interpretation is efficient in analyzing classical proofs because it supports Markov’s p... |

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Citation Context ...le of omniscience and weak König’s lemma. We discuss some applications, as well as some previous results in the literature. 1 Introduction The realizability method, invented by Stephen Kleene in 1945 =-=[8]-=-, is a way of making explicit the constructive content of arithmetical sentences. It is closely related with intuitionistic logic, being reminiscent of the BrouwerHeyting-Kolmogorov interpretation of ... |

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Citation Context ...ype 0 � (0 � 0) and z of type 0 (the constant z is needed to ensure that each finite type is inhabited by at least one closed term). The theory IL ω ≤ is intuitionistic logic in all finite types (see =-=[2]-=- for the Hilbert-type deduction system that we use) with axioms stating that ≤0 is reflexive, transitive, and with the axioms A1 : x ≤0 m(x, y) ∧ y ≤0 m(x, y) A2 : x ≤0 x ′ ∧ y ≤0 y ′ → m(x, y) ≤0 m(x... |

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Citation Context ...on functional M of type 1 � 1 satisfying the equations Mf0 =0 f0 and Mf(n + 1) =0 max0(Mfn, f(n + 1)). We usually write f M instead of Mf. Note that f ≤∗ 1 f M . The following result is due to Howard =-=[7]-=- for HA ω , and to Kohlenbach [9] for the other theories: 15sProposition 6. The theories HA ω , PRA ω and GnA ω i (n ≥ 2) are majorizability theories. The soundness theorem has an arithmetical extensi... |

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Citation Context ...a quantifier-free formula. It is known that the type 1 closed terms of the language of PRA ω i define the primitive recursive functions. The theories GnAω i (n ≥ 2) were introduced by Kohlenbach (see =-=[9]-=- for details). They form a sequence of increasing strength, closely related to the levels of Grzegorczyk’s hierarchy of primitive recursive functions. The principle of induction present is the restric... |

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Citation Context ...der. † Partially supported by pocti/fct and feder. 1sfor a modern overview of realizability). In particular, Kleene’s realizability and related notions (e.g., modified realizability due to Kreisel in =-=[13]-=-) decides disjunctions and provides precise witnesses for existential statements. In this paper, we shift attention from the constructive content of arithmetical sentences to the program of extracting... |

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Citation Context ...m (semiintuitionistic) proofs in arithmetic. The emphasis on bounds is, by no means, a novel idea: Ulrich Kohlenbach has, for some years now, been urging a shift of attention to numerical bounds (cf. =-=[12]-=- for a recent statement). Bounded modified realizability, as we call the novel realizability notion, is based on an analysis of formulas which always disregards precise existential witnesses and decis... |

16 |
Varieties of constructive mathematics, volume 97
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Citation Context ...nd the types of x and y are unrestricted, is benign. Note that when both types are 0 and A and B are quantifier-free formulas, we have the well-known lesser limited principle of omniscience LLPO (cf. =-=[4]-=-). Also, by the argument after Corollary 3, it follows from (§) that the scheme ∀x∀y(¬A(x) ∨ ¬B(y)) → ∀x¬A(x) ∨ ∀y¬B(y), where A and B are arbitrary, is benign. In order to see that (§) is benign, obs... |

13 |
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(Show Context)
Citation Context ...ent from the monotone realizability introduced by Kohlenbach in [10], which is still a notion based on ordinary (modified) realizability. Our new form of realizability is inspired by work realized in =-=[5]-=- on a newly found functional interpretation (in the sense of Gödel [6]), but it is simpler than it in one crucial aspect. The new functional interpretation relies on intensional majorizability relatio... |

13 | Relative Constructivity
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(Show Context)
Citation Context ...las which always disregards precise existential witnesses and decisions concerning disjunctive statements. In this respect, it is different from the monotone realizability introduced by Kohlenbach in =-=[10]-=-, which is still a notion based on ordinary (modified) realizability. Our new form of realizability is inspired by work realized in [5] on a newly found functional interpretation (in the sense of Göde... |

10 |
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(Show Context)
Citation Context ... In view of the fact that T ω +bAC ω +bIP ω ˜ ∃free +MAJ ω refutes Markov’s principle, one is drawn to the conclusion that Π 0 1 − LEM does not prove Markov’s principle. This result first appeared in =-=[1]-=-. As in the discussion of the previous point, by the argument following Corollary 3, we get that the scheme ¬A∨¬¬A is benign, where A is arbitrary. 4. The following principle can easily be put in Πbr-... |

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(Show Context)
Citation Context ..., k) → ∃x ≤1 u ∀k A(x, k)), can be put in the form Πbr when A is a quantifier-free formula. Therefore, it is a benign principle. A related principle is uniform weak König’s lemma (first considered in =-=[11]-=-), UWKL for short: ∃Φ 1�1 ∀f 1 (∀k∃s ∈ {0, 1} k (s ∈ f tree (s)) → ∀k (Φf(k) ∈ f tree )), where we are using the notation of [2]. Remark that the functional Φ1�1 can be bounded (in the sense of ≤∗ 1�1... |

6 |
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Citation Context ...unction, implication, bounded quantifications and monotone universal quantifications, i.e., quantifications of the form ˜ ∀a(. . .). 5sThis notion reminds the well-known notion of ∃-free formula (cf. =-=[16]-=-), but do observe that it allows disjunctions. We now proceed with the definition of the new realizability notion. We chose to define it in a slightly unfamiliar way. Instead of saying what are realiz... |

4 |
Introductory note to 1958 and 1972
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(Show Context)
Citation Context ...see 1 of Subsection 4.3). Due to this feature, we opt for a treatment of equality based on the minimal alternative described by Troelstra in the end of section 3.1 and the beginning of section 3.3 of =-=[14]-=-. In the minimal alternative, there is a symbol of equality for terms of type 0 only. Its axioms are E1 : x =0 x E2 : x =0 y ∧ φ[x/w] → φ[y/w] where φ is an atomic formula with a distinguished type 0 ... |