The analytic Markov’s principle states that the pseudo-order on the Dedekind real numbers is a stable relation: for all real numbers and , implies .
This is equivalent to the usual formulation of the analytic Markov’s principle, which says that for all real numbers , implies . For if we take , this becomes implies , and by the order and arithmetic properties of the real numbers, this is equivalent to implies , which is the same as implies .
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 -algebra as well, where the relation is in general only a strict weak order instead of a pseudo-order, the preorder is not a partial order, and the equivalence relation derived from the preorder holds if and only if is nilpotent. The quotient of the Weil -algebra by its nilradical is the Dedekind real numbers satisfying the analytic Markov’s principle.
Last revised on January 18, 2024 at 01:03:07. See the history of this page for a list of all contributions to it.