Reverse mathematics and Isbell's zig‐zag theorem

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.