Reverse mathematics and marriage problems with unique solutions

We analyze the logical strength of theorems on marriage problems with unique solutions using the techniques of reverse mathematics, restricting our attention to problems in which each boy knows only finitely many girls. In general, these marriage theorems assert that if a marriage problem has a unique solution then there is a way to enumerate the boys so that for every m, the first m boys know exactly m girls. The strength of each theorem depends on whether the underlying marriage problem is finite, infinite, or bounded. Our goal is to analyze the logical strength of some marriage theorems using the framework of reverse mathematics. The subsystems of second order arithmetic used here are RCA0, which includes comprehension for recursive (also called computable) sets of natural numbers, WKL0, which appends a weak form of König’s Lemma for trees, and ACA0, which appends comprehension for arithmetically definable sets. Simpson’s book [5] provides detailed axiomatizations of the subsystems and extensive development of the program of reverse mathematics. ∗Authors’ address: Department of Mathematical Sciences, Appalachian State University, Boone, NC 28608 Corresponding author: Jeffry L. Hirst [email protected] TEL: 1-828-262-2861 FAX: 1-828-265-8617