Reverse Mathematics

Reverse mathematics is a new take on an old idea: asking which axioms are necessary to prove a given theorem. This question was first asked about the parallel axiom in Euclid’s geometry and later about the axiom of choice in set theory. Obviously, such questions can be asked in many fields of mathematics, but in recent decades, it has proved fruitful to focus on subsystems of second-order arithmetic, where much of mainstream mathematics resides. It has been found that many basic theorems of analysis and topology, as well as certain parts of infinite algebra and combinatorics, can be proved in such systems. And, remarkably, almost all the basic theorems fall into one of five particular systems: a base system RCA0 of “constructive mathematics” and four others defined by certain axioms about real numbers. Moreover, many of the theorems not provable in RCA0 turn out to be equivalent to one of these defining axioms, so we know precisely which axiom is needed to prove them. Thus, after some motivational remarks about the parallel axiom and the axiom of choice, we will concentrate on the study of RCA0 and its extensions, which is what “reverse mathematics” is generally taken to mean today.