## On weak Markov's principle (2002)

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Venue: | MLQ MATH. LOG. Q |

Citations: | 4 - 1 self |

### BibTeX

@ARTICLE{Kohlenbach02onweak,

author = {Ulrich Kohlenbach},

title = {On weak Markov's principle},

journal = {MLQ MATH. LOG. Q},

year = {2002},

volume = {48},

pages = {59--65}

}

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### Abstract

We show that the so-called weak Markov's principle (WMP) which states that every pseudo-positive real number is positive is underivable in T # :=EHA # +AC. Since T # allows to formalize (at least large parts of) Bishop's constructive mathematics this makes it unlikely that WMP can be proved within the framework of Bishop-style mathematics (which has been open for about 20 years). The underivability even holds if the ine#ective schema of full comprehension (in all types) for negated formulas (in particular for #-free formulas) is added which allows to derive the law of excluded middle for such formulas.

### Citations

430 | Constructive Analysis
- Bishop, Bridges
- 1985
(Show Context)
Citation Context ...nvestigated by Ishihara ([8],[9]). WMP plays a crucial role in the study of the interrelations between various continuity principles within the framework of Bishop-style constructive mathematics ([2],=-=[3]-=-,[4]). In order to state WMP we first need the notion of ‘pseudo-positivity’: Definition 1 1) A real number a ∈ IR is pseudo-positive if 2) a ∈ IR is positive if a > 0. ∀x ∈ IR(¬¬(0 < x) ∨ ¬¬(x < a)).... |

274 |
Foundations of Constructive Mathematics
- Beeson
- 1985
(Show Context)
Citation Context ...ositive if 2) a ∈ IR is positive if a > 0. ∀x ∈ IR(¬¬(0 < x) ∨ ¬¬(x < a)). Remark 2 1) It is clear that we can without loss of generality restrict x ∈ IR in the definition of pseudo-positivity to x ∈ =-=[0, 1]-=-. 2) ‘x > y’ for x, y ∈ IR is to be read as a positive existence statement ‘∃n ∈ IN(x ≥ y + 1 )’ which has – constructively – to be distinguished from n+1 the negative statement ‘¬(x ≤ y)’. Definition... |

126 |
Metamathematical Investigation of Intuitionistic Arithmetic and Analysis
- Troelstra
- 1973
(Show Context)
Citation Context ...f course, not completely precise as no particular formal system has been identified with Bishop-style mathematics. However, it is commonly agreed that Heyting arithmetic in all finite types HA ω (see =-=[17]-=-) plus the axiom of choice AC in all types AC ρ,τ : ∀x ρ ∃y τ A(x, y) → ∃Y ρ→τ ∀x ρ A(x, Y (x)) is a framework which is quite capable of formalizing existing constructive (in the sense of Bishop) math... |

59 |
Varieties of constructive mathematics
- Bridges, Richman
- 1987
(Show Context)
Citation Context ...tigated by Ishihara ([8],[9]). WMP plays a crucial role in the study of the interrelations between various continuity principles within the framework of Bishop-style constructive mathematics ([2],[3],=-=[4]-=-). In order to state WMP we first need the notion of ‘pseudo-positivity’: Definition 1 1) A real number a ∈ IR is pseudo-positive if 2) a ∈ IR is positive if a > 0. ∀x ∈ IR(¬¬(0 < x) ∨ ¬¬(x < a)). Rem... |

43 |
Hereditarily majorizable functionals of finite type
- Howard
- 1973
(Show Context)
Citation Context ...ω witnessing ‘∃x’ uniformly in F, f, g. However, such a term would satisfy the Gödel functional (‘Dialectica’) interpretation of the extensionality axiom for functionals of type 2. As shown by Howard =-=[7]-=- such a term does not exist in E-HA ω as it would not be majorizable whereas all closed terms in E-HA ω are. 4 Using our theorem 5 instead we can also extend this proof to the situation where CA¬ is a... |

16 |
Continuity properties in constructive mathematics
- Ishihara
- 1992
(Show Context)
Citation Context ...l Research Foundation. 1(ASP) and in the latter as ‘weak limited principle of existence’ (WLPE)). Under the currently common name of weak Markov’s principle it has been investigated by Ishihara ([8],=-=[9]-=-). WMP plays a crucial role in the study of the interrelations between various continuity principles within the framework of Bishop-style constructive mathematics ([2],[3],[4]). In order to state WMP ... |

15 |
Continuity and nondiscontinuity in constructive mathematics
- Ishihara
- 1991
(Show Context)
Citation Context ...ional Research Foundation. 1(ASP) and in the latter as ‘weak limited principle of existence’ (WLPE)). Under the currently common name of weak Markov’s principle it has been investigated by Ishihara (=-=[8]-=-,[9]). WMP plays a crucial role in the study of the interrelations between various continuity principles within the framework of Bishop-style constructive mathematics ([2],[3],[4]). In order to state ... |

13 | Relative Constructivity
- Kohlenbach
- 1998
(Show Context)
Citation Context ...ble. 32 The independence result Definition 4 (Independence-of-premise for negated formulas) IP¬ : (¬A → ∃x ρ B) → ∃x ρ (¬A → B), where x does not occur free in A. The following theorem was proved in =-=[12]-=-(thm.3.3) Theorem 5 ([12]) Let δ, ρ, γ be arbitrary types and G be a sentence of the form G ≡ ∀x δ (A → ∃y ≤ρ sx¬B(x, y)) and ˜G :≡ ∃Y ≤ s∀x(A → ¬B(x, Y (x))), where s is a closed term of E-HA ω and x... |

10 | On uniform weak König’s lemma - Kohlenbach - 2002 |

7 |
Markov’s principle, church’s thesis and lindelöf’s theorem
- Ishihara
- 1993
(Show Context)
Citation Context ...do-positive real number is positive. WMP follows easily from the well-known Markov’s principle as well as from an appropriate continuity principle and also from the extended Church’s thesis ECT0 (see =-=[10]-=-). So WMP holds both in Russian constructive mathematics as well as in intuitionistic mathematics (in the sense of [4]). Since about 20 years it has been an open problem whether WMP is derivable in Bi... |

6 | Finite type arithmetic: Computable existence analysed by modified realisability and functional interpretation - Jørgensen - 2001 |

2 |
J.,The formalization of Bishop’s constructive mathematics
- Goodman, Myhill
- 1972
(Show Context)
Citation Context ...oice AC in all types AC ρ,τ : ∀x ρ ∃y τ A(x, y) → ∃Y ρ→τ ∀x ρ A(x, Y (x)) is a framework which is quite capable of formalizing existing constructive (in the sense of Bishop) mathematics (see also [1],=-=[6]-=-). In this note we show that WMP is underivable even in E-HA ω +AC, where E-HA ω is Heyting arithmetic in all finite types with the full axiom of extensionality (see again 2[17] for a precise definit... |

2 | Markov's principle, Chruch's thesis and Lindelof's theorem - Ishihara - 1993 |

2 | Finite Type Arithmetic: Computable Existence Analysed by Modi Realisability - Jrgensen |

2 | Constructive continuity. Mem.Amer.Math.Soc. 277 - Mandelkern - 1983 |

2 |
Constructive complete finite sets
- Mandelkern
- 1988
(Show Context)
Citation Context ...formulas) is added, which allows one to derive the law of excluded middle for such formulas. 1 Introduction The so-called weak Markov’s principle (WMP) has been first considered by Mandelkern in [14],=-=[15]-=- (in the former paper under the name ‘almost separating principle’ ∗ Basic Research in Computer Science, funded by the Danish National Research Foundation. 1(ASP) and in the latter as ‘weak limited p... |

2 |
Weak Markov’s principle, strong extensionality, and countable choice
- Richman
- 2000
(Show Context)
Citation Context ...ere the underivability of the usual formulation of WMP in E-HA ω +AC we would have to undertake the tedious task of verifying that Ishihara’s equivalence proof can be formalized in EHA ω +AC. Richman =-=[16]-=- has shown that the proof that WMP implies Ishihara’s strong extensionality statement requires a weak form of countable choice and fails in certain sheaf models. 1 Note that E-HA ω only has prime form... |

2 |
On the uniform weak König’s lemma. Ann. Pure Applied Logic 114
- Kohlenbach
- 2002
(Show Context)
Citation Context ... classical binary König’s lemma WKL (and even the uniform binary König’s lemma UWKL which states the existence of a functional which selects an infinite path uniformly in an infinite binary tree, see =-=[13]-=-). Many equivalent formulations of WMP have been found meanwhile. One of those, due to Ishihara [8], is particularly interesting and reads as follows Every mapping of a complete metric space into a me... |