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To be concrete: Let's define Markov's Principle as $$\forall P \subseteq \mathbb N, (\forall n \in \mathbb N, n \in P \vee n \notin P) \to \neg(\forall n \in \mathbb N, n \notin P) \to \exists n \in \mathbb N, n \in P$$ and the Fan Theorem as stated on its nLab page.

sarahzrf
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  • Both seem obviously true. If I was Markov I would prefer to be remembered for my work on probability chains. – Michael Sep 17 '18 at 19:27
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    @Michael Markov's principle isn't obviously true (indeed, it's not true) in intuitionistic logic. To use Curry-Howard in Martin-Löf intensional type theory: let $P$ be the set ${0}$ if the twin prime conjecture is provably true, ${1}$ if the twin prime conjecture is provably false, and ${2}$ if it's neither. Then $\exists n \in \mathbb{N} : n \in P$ isn't currently known to be an inhabited type, so the implication that is Markov's principle isn't known to be true when specialised to that particular $P$. – Patrick Stevens Sep 17 '18 at 19:54
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    @Michael Also, Russian constructivism Markov was Markov chain Markov’s son. – spaceisdarkgreen Sep 17 '18 at 20:14
  • @PatrickStevens : Then intuitionist logic seems counter-intuitive! – Michael Sep 17 '18 at 20:23
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    @Michael Those are two different Markov's, the propability theory guy is the father, the computability theory guy is the son. Also, I motive the constructive logics here so that, under this reading, it hopefully doesn't feel as counter-intuitive. And sarahzrf, In the description section of the video you find a discussions about equivalences of potentially non-constructive principles in great detail, in particularly this 200 page thesis on the arXiv. – Nikolaj-K Jan 03 '20 at 17:18

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To show that Markov's principle doesn't imply the fan theorem:

Markov's principle holds in the effective topos. However, the fan theorem is false in the effective topos, using (the complement of) the Kleene tree. These are respectively corollary 3.1.4 and theorem 3.2.25 in Van Oosten, Realizability: An Introduction to its Categorical Side.

To show that the fan theorem doesn't imply Markov's principle:

In sheaves over Cantor space the fan theorem holds, but not Markov's principle. The fan theorem holds in any topos of sheaves over a topological space (theorem 3.2 in Fourman & Hyland, Sheaf Models for Analysis). To show Markov's principle fails, let $\alpha$ be the generic element of Cantor space. Then the truth value of $(\exists n) \;\alpha(n) = 1$ is $2^\mathbb{N} \setminus \{\lambda x.0\}$, but the truth value of $(\forall n) \;\alpha(n) = 0$ is the interior of $\{\lambda x.0\}$, which is empty, and so the truth value of $\neg (\forall n) \;\alpha(n) = 0$ is $2^\mathbb{N}$ and in particular contains the constant function $\lambda x.0$, so the truth value of $\neg (\forall n) \;\alpha(n) = 0 \;\rightarrow\;(\exists n) \;\alpha(n) = 1$ is not all of $2^\mathbb{N}$.

aws
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