## 2001), Intuitionistic choice and restricted classical logic

Venue: | Mathematical Logic Quarterly |

Citations: | 7 - 1 self |

### BibTeX

@ARTICLE{Kohlenbach_2001),intuitionistic,

author = {Ulrich Kohlenbach},

title = {2001), Intuitionistic choice and restricted classical logic},

journal = {Mathematical Logic Quarterly},

year = {},

pages = {455--460}

}

### OpenURL

### Abstract

König’s lemma, primitive recursive arithmetic.

### Citations

127 |
Metamathematical investigation of intuitionistic Arithmetic andAnalysis
- Troelstra
- 1973
(Show Context)
Citation Context ...wise, we get a characteristic function for B(n). So by applying LLOP to f,g we obtain LNOS. ✷ In the following, M ω ,IP ω 0 denote the Markov principle resp. the independence-ofpremise principle from =-=[11]-=-(3.5.10). Theorem 2 1) HA ω +AC+M ω +IP ω 0 +LNOS is Π 0 2-conservative over HA. 2) � HA ω +AC+Mω +IPω 0 +LNOS is Π02 -conservative over PRA. If AC is replaced by AC0,τ plus AC! 1,τ (with arbitrary τ)... |

81 | Gödel’s functional (“Dialectica”) interpretation
- Avigad, Feferman
(Show Context)
Citation Context ...ic strength of such systems can be determined by functional interpretation based using his non-constructive µ-operator and his classical results on the strength of systems based on this operator (see =-=[1]-=- for a survey of those results). In this note we show that a similar use of functional interpretation combined with the majorization arguments which we developed in [8] can be used to determine the st... |

60 |
Constructive Mathematics
- Bishop, Bridges
- 1985
(Show Context)
Citation Context ...f lesser numerical omnisience ⎧ ⎪⎨ ∀n LNOS :≡ ⎪⎩ 0 ((A(n) ∨¬A(n)) ∧ (B(n) ∨¬B(n)))∧ ¬(∃nA(n) ∧∃nB(n)) →∀n¬A(n)∨∀n¬B(n), which generalizes the well-known ‘lesser limited principle of omniscience’ (see =-=[2]-=- for various equivalent formulations of this principle) LLOP :≡ ∀f 0 ,g 0 (¬(∃n(fn =0)∧∃n(gn =0))→∀n(fn �=0)∨∀n(gn �= 0)) inthesamewayasNOS generalizes LPO. We will define a system based on LNOS and t... |

47 | Theories of finite type related to mathematical practice - Feferman - 1977 |

46 |
Hereditarily majorizable functionals of finite type
- Howard
(Show Context)
Citation Context ...1 → 0), ≤ρ is defined pointwise and 1 ρ := λf, g.1. By functional interpretation (see [11](3.5.10)) one extracts a closed term Φ of HA ω such that HA ω ⊢∀x∀F ≤1(∀f,g A0(f, g, Ffg, ΦFx)→∃yR(x, y)). By =-=[7]-=-, Φ has a majorizing functional Φ ∗ and hence (using basic properties of majorization in Howard’s sense) Put together we get and hence HA ω ⊢∀x∀F ≤1(tx := Φ ∗ 1 ρ x ≥ ΦFx). HA ω ⊢∀x∀F ≤1(∀f,g∀z ≤ txA0... |

39 |
Effective bounds from ineffective proofs in analysis: an application of functional interpretation and majorization
- Kohlenbach
- 1992
(Show Context)
Citation Context ...ems based on this operator (see [1] for a survey of those results). In this note we show that a similar use of functional interpretation combined with the majorization arguments which we developed in =-=[8]-=- can be used to determine the strength of systems which instead of NOS are based on the weaker schema of lesser numerical omnisience ⎧ ⎪⎨ ∀n LNOS :≡ ⎪⎩ 0 ((A(n) ∨¬A(n)) ∧ (B(n) ∨¬B(n)))∧ ¬(∃nA(n) ∧∃nB... |

29 |
Extensional Gödel functional interpretation—a consistency proof of classical analysis
- Luckhardt
- 1973
(Show Context)
Citation Context ... has a functional interpretation in � HA ω | \ which is Π0 2- conservative over PRA. The claim for the fully extensional systems follows by the well-known elimination of extensionality technique (see =-=[10]-=- for details). ✷ In [6] an extension of the usual weak König’s lemma WKL to binary trees given by arbitrary formulas Φ(x,m) which are decidable in the variable m which defines the tree, i.e. ∀m(Φ(x,m)... |

17 | Intuitionistic choice and classical logic
- Coquand, Palmgren
- 2000
(Show Context)
Citation Context ...this fact relies on functional interpretation and a majorization technique developed in a previous paper. ∗ Basic Research in Computer Science, Centre of the Danish National Research Foundation. 1sIn =-=[6]-=-, systems of intuitionistic arithmetic in all finite types extended by various kinds of the axiom of choice and the schema of numerical omnisience NOS: ∀n(A(n) ∨¬A(n)) →∀nA(n)∨∃n¬A(n), where n ranges ... |

13 | Relative constructivity
- Kohlenbach
- 1998
(Show Context)
Citation Context ...re applied aspect of extracting algorithms or bounds from proofs of semi-classical systems, then (at least in the absence of M ω ) 4 4 For a strong result in this direction in the presence of M ω see =-=[9]-=-(thm.3.18). 7sthe much stronger results can be obtained as we have shown in [9]. E.g. consider the comprehension principle for negated formulas in all types CA¬ : ∃Φ ρ→0 ∀x ρ (Φ(x) =0↔¬A(x)) (where A ... |

12 | E#ective bounds from ine#ective proofs in analysis: an application of functional interpretation and majorization - Kohlenbach - 1992 |

10 |
Note on the fan theorem
- Troelstra
- 1974
(Show Context)
Citation Context ... We show the theorem for � HA ω | \+AC0,0 +LNOS. Analogously to the proof of the lemma above one verifies that � HA ω | \+AC0,0 allows to reduce DWKL to the usual weak König’s lemma WKL as defined in =-=[12]-=-: WKL:≡ ∀f 1 (T(f)∧∀x 0 ∃n 0 (lth(n) =x∧fn =0)→∃b 1 ∀x 0 (f(bx) = 0)), where Tf :≡∀n 0 ,m 0 (f(n∗m)=0 0→fn =0 0) ∧∀n 0 ,x 0 (f(n∗〈x〉)=0 0→x≤0 1). Consider the formula3 ⎧ ⎪⎨ (+) ⎪⎩ →∃m≤1(k −· 1)(lth(m)... |

3 |
Strict Constructivism and The Philosophy of Mathematics
- Ye
- 2000
(Show Context)
Citation Context ...nomial bounds are guaranteed. Remark 6 Intuitionistically one can allow certain induction principles which classically would go beyond the strength of PRA and still obtain conservation over PRA. E.g. =-=[13]-=- considered function parameter free forms of induction rules for fomulas like ∃f 1 ∀x 0 A0 (with quantifier-free A0). It seems likely that also in this context one may add LNOS and still preserve PRA-... |

1 |
On the proof theoretical strength of some systems with the numerical omniscience scheme (abstract
- Feferman
- 2000
(Show Context)
Citation Context ...ed by various kinds of the axiom of choice and the schema of numerical omnisience NOS: ∀n(A(n) ∨¬A(n)) →∀nA(n)∨∃n¬A(n), where n ranges over the natural numbers and A is any formula1 , are studied. In =-=[5]-=-, Feferman noticed that the proof theoretic strength of such systems can be determined by functional interpretation based using his non-constructive µ-operator and his classical results on the strengt... |