Introduction to reverse math [pdf]
In math we typically assume a set of axioms to prove a theorem. In reverse mathematics, the premise is reversed: we start with a theorem and try to determine the minimal axiomatic system required to prove the theorem (over a weak base system). This produces interesting results, as it can be shown that theorems from different fields of math such as group theory and analysis are in fact equivalent. Also, using reverse mathematics we can put theorems into a hierarchy by their complexity such that theorems that can be proven with weaker subsystems are “less complex”.
Some exercises from Robert Soare’s amazing book on computability theory, “Computability Theory and Applications”.
Computability Theory Proofs [pdf]