Markov's principle, named after Andrey Markov Jr, is a statement in computability theory that  is logical validity classically  not in intuitionistic constructive mathematics. Many particular instances of the principle are nevertheless provable there as well. There are many equivalent formulations of Markov's principle, as listed below. 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 constructive analysis It is equivalent, in the language of real analysis, to the following principles: Realizability If constructive arithmetic is translated using realizability into a classical meta-theory that proves the -consistency of the relevant classical theory (for example, Peano Arithmetic if we are studying Heyting arithmetic), then Markov's principle is justified: a realizer is the constant function that takes a realization that is not everywhere false to the unbounded search that successively checks if is true. If is not everywhere false, then by -consistency there must be a term for which holds, and each term will be checked by the search eventually. If however does not hold anywhere, then the domain of the constant function must be empty, so although the search does not halt it still holds vacuously that the function is a realizer. By the Law of the Excluded Middle (in our classical metatheory), must either hold nowhere or not hold nowhere, therefore this constant function is a realizer. If instead the realizability interpretation is used in a constructive meta-theory, then it is not justified. Indeed, for first-order arithmetic, Markov's principle exactly captures the difference between a constructive and classical meta-theory. Specifically, a statement is provable in Heyting arithmetic with Extended Church's thesis if and only if there is a number that provably realizes it in Heyting arithmetic; and it is provable in Heyting arithmetic with Extended Church's thesis and Markov's principle if and only if there is a number that provably realizes it in Peano arithmetic .
Markov's principle, named after Andrey Markov Jr, is a conditional existence statement for which there are many equivalent formulations, as discussed below. The principle is logical validity classically , but not in intuitionistic constructive mathematics. However, many particular instances of it are nevertheless provable in a constructive context as well. 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. Realizability If constructive arithmetic is translated using realizability into a classical meta-theory that proves the -consistency of the relevant classical theory (for example, Peano Arithmetic if we are studying Heyting arithmetic), then Markov's principle is justified: a realizer is the constant function that takes a realization that is not everywhere false to the unbounded search that successively checks if is true. If is not everywhere false, then by -consistency there must be a term for which holds, and each term will be checked by the search eventually. If however does not hold anywhere, then the domain of the constant function must be empty, so although the search does not halt it still holds vacuously that the function is a realizer. By the Law of the Excluded Middle (in our classical metatheory), must either hold nowhere or not hold nowhere, therefore this constant function is a realizer. If instead the realizability interpretation is used in a constructive meta-theory, then it is not justified. Indeed, for first-order arithmetic, Markov's principle exactly captures the difference between a constructive and classical meta-theory. Specifically, a statement is provable in Heyting arithmetic with Extended Church's thesis if and only if there is a number that provably realizes it in Heyting arithmetic; and it is provable in Heyting arithmetic with Extended Church's thesis and Markov's principle if and only if there is a number that provably realizes it in Peano arithmetic. In constructive analysis It is equivalent, in the language of real analysis, to the following principles: 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 .
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statement in computability theory that
conditional existence statement for which there are many equivalent formulations, as discussed below.
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The principle
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the principle
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in a constructive context
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There are many equivalent formulations of Markov's principle, as listed below.
A
Realizability If constructive arithmetic is translated using realizability into a classical meta-theory that proves the -consistency of the relevant classical theory (for example, Peano Arithmetic if we are studying Heyting arithmetic), then Markov's principle is justified: a realizer is the constant function that takes a realization that is not everywhere false to the unbounded search that successively checks if is true. If is not everywhere false, then by -consistency there must be a term for which holds, and each term will be checked by the search eventually. If however does not hold anywhere, then the domain of the constant function must be empty, so although the search does not halt it still holds vacuously that the function is a realizer. By the Law of the Excluded Middle (in our classical metatheory), must either hold nowhere or not hold nowhere, therefore this constant function is a realizer.
A
If instead the realizability interpretation is used in a constructive meta-theory, then it is not justified. Indeed, for first-order arithmetic, Markov's principle exactly captures the difference between a constructive and classical meta-theory. Specifically, a statement is provable in Heyting arithmetic with Extended Church's thesis if and only if there is a number that provably realizes it in Heyting arithmetic; and it is provable in Heyting arithmetic with Extended Church's thesis and Markov's principle if and only if there is a number that provably realizes it in Peano arithmetic.
D
Realizability If constructive arithmetic is translated using realizability into a classical meta-theory that proves the -consistency of the relevant classical theory (for example, Peano Arithmetic if we are studying Heyting arithmetic), then Markov's principle is justified: a realizer is the constant function that takes a realization that is not everywhere false to the unbounded search that successively checks if is true. If is not everywhere false, then by -consistency there must be a term for which holds, and each term will be checked by the search eventually. If however does not hold anywhere, then the domain of the constant function must be empty, so although the search does not halt it still holds vacuously that the function is a realizer. By the Law of the Excluded Middle (in our classical metatheory), must either hold nowhere or not hold nowhere, therefore this constant function is a realizer.
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If instead the realizability interpretation is used in a constructive meta-theory, then it is not justified. Indeed, for first-order arithmetic, Markov's principle exactly captures the difference between a constructive and classical meta-theory. Specifically, a statement is provable in Heyting arithmetic with Extended Church's thesis if and only if there is a number that provably realizes it in Heyting arithmetic; and it is provable in Heyting arithmetic with Extended Church's thesis and Markov's principle if and only if there is a number that provably realizes it in Peano arithmetic
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