The paper explores the logical strength of Isbell's zig‐zag theorem using the framework of reverse mathematics. Working in, we show that is equivalent to Isbell's zig‐zag theorem for countable monoids: If B is a monoid extension of A, then is dominated by A if and only if b has a zig‐zag over A. Our proof of Isbell's zig‐zag theorem avoids use of strong comprehension axioms common in traditional proofs. We also analyze the strength of theorems concerning binary relations.