 In computability theory In the language of computability theory, Markov's principle is a formal expression of the claim that if it is impossible that an algorithm does not terminate, then it does terminate. This is equivalent to the claim that if a set and its complement are both computably enumerable, then the set is decidable. In intuitionistic logic A predicate P over some domain is called decidable if for every x in the domain, either P(x) is true, or P(x) is not true , which is not always the case constructively. In predicate logic, if P is a predicate  over the natural numbers, it is expressed as : That is, if P is decidable, and it cannot be false for every natural number n, then it is true for some n. This form can be justified by Brouwer's continuity principles, whereas the stronger form contradicts them. Thus it can be derived from intuitionistic, realizability, and classical reasoning, in each case for different reasons, but this principle is not valid in the general constructive sense of Bishop.Ulrich Kohlenbach, "On weak Markov's principle". Mathematical Logic Quarterly (2002), vol 48, issue S1, pp. 59–65. History The principle was first studied and adopted by the Russian school of constructivism, together with choice principles and often together with a realizability perspective on the notion of mathematical function.
History The principle was first studied and adopted by the Russian school of constructivism, together with choice principles and often with a realizability perspective on the notion of mathematical function. In computability theory In the language of computability theory, Markov's principle is a formal expression of the claim that if it is impossible that an algorithm does not terminate, then it does terminate. This is equivalent to the claim that if a set and its complement are both computably enumerable, then the set is decidable. In Intuitionistic logic In predicate logic, a predicate P over some domain is called decidable if for every x in the domain, either P(x) is true, or P(x) is not true . This is not trivially true constructively. For a decidable predicate P over the natural numbers, Markov's principle then reads : That is, if P cannot be false for all natural numbers n, then it is true for some n. This form can be justified by Brouwer's continuity principles, whereas the stronger form contradicts them. Thus it can be derived from intuitionistic, realizability, and classical reasoning, in each case for different reasons, but this principle is not valid in the general constructive sense of Bishop.Ulrich Kohlenbach, "On weak Markov's principle". Mathematical Logic Quarterly (2002), vol 48, issue S1, pp. 59–65. 
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History The principle was first studied and adopted by the Russian school of constructivism, together with choice principles and often with a realizability perspective on the notion of mathematical function.
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intuitionistic logic A predicate
Intuitionistic logic In predicate logic, a predicate
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, which is not always the case constructively. In predicate logic, if P is a predicate
. This is not trivially true constructively.
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For a decidable predicate P
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it is expressed as
Markov's principle then reads
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is decidable, and it
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every natural number
all natural numbers
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History The principle was first studied and adopted by the Russian school of constructivism, together with choice principles and often together with a realizability perspective on the notion of mathematical function.