   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.
<coherence>    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.
 In constructive analysis It is equivalent, in the language of real analysis, to the following principles: 