Markov's principle

Markov's rule is the formulation of Markov's principle as a rule. It states that ${\displaystyle \exists n\;P(n)}$ is derivable as soon as ${\displaystyle \neg \neg \exists n\;P(n)}$ is, for ${\displaystyle P}$ decidable. Formally,

${\displaystyle \forall n(P(n)\lor \neg P(n)),\ \neg \neg \exists n\;P(n)\ \vdash \ \exists n\;P(n)}$

Anne S. Troelstra[1] proves that it is an admissible rule in Heyting arithmetic. Later, the logician Harvey Friedman showed that Markov's rule is an admissible rule in all of intuitionistic logic, Heyting arithmetic, and various other intuitionistic theories,[2] using the Friedman translation.

If constructive arithmetic is translated using realizability into a classical meta-theory that proves the ${\displaystyle \omega }$-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 ${\displaystyle P}$ is not everywhere false to the unbounded search that successively checks if ${\displaystyle P(0),P(1),P(2),\dots }$ is true. If ${\displaystyle P}$ is not everywhere false, then by ${\displaystyle \omega }$-consistency there must be a term for which ${\displaystyle P}$ holds, and each term will be checked by the search eventually. If however ${\displaystyle P}$ 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), ${\displaystyle P}$ 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.

The Weak Markov's principle is a weaker form of Markov's principle may be stated in the language of analysis as

${\displaystyle \forall x\in \mathbb {R} \ \ (\forall y\in \mathbb {R} \ \ \neg \neg (0<y)\vee \neg \neg (y<x))\to 0<x.}$

It is a conditional statement about the decidability of positivity of a real number.

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.[3]