nLab analytic Markov's principle




The analytic Markov’s principle states that the pseudo-order on the Dedekind real numbers is a stable relation: for all real numbers rr \in \mathbb{R} and ss \in \mathbb{R}, ¬¬(r<s)\neg \neg (r \lt s) implies r<sr \lt s.

This is equivalent to the usual formulation of the analytic Markov’s principle, which says that for all real numbers xx \in \mathbb{R}, ¬(x0)\neg (x \leq 0) implies 0<x0 \lt x. For if we take x=srx = s - r, this becomes ¬(sr0)\neg (s - r \leq 0) implies 0<sr0 \lt s - r, and by the order and arithmetic properties of the real numbers, this is equivalent to ¬(sr)\neg (s \leq r) implies r<sr \lt s, which is the same as ¬¬(r<s)\neg \neg (r \lt s) implies r<sr \lt s.

Other equivalent statements include that the tight apartness relation on the Dedekind real numbers is a stable relation.

The analytic Markov’s principle makes sense for any ordered local Artinian \mathbb{R} -algebra as well, where the relation <\lt is in general only a strict weak order instead of a pseudo-order, the preorder \geq is not a partial order, and the equivalence relation aba \approx b derived from the preorder holds if and only if aba - b is nilpotent. The quotient of the Weil \mathbb{R}-algebra by its nilradical is the Dedekind real numbers satisfying the analytic Markov’s principle.

See also


Last revised on January 18, 2024 at 01:03:07. See the history of this page for a list of all contributions to it.