On sequential theorems in Reverse Mathematics

Abstract

Many theorems of mathematics have the form that for a certain problem, e.g. a differential equation or polynomial (in)equality, there exists a solution. The sequential version then states that for a sequence of problems, there is a sequence of solutions. The original and sequential theorem can often be proved via the same (or similar) proof and often have the same (or similar) logical properties, esp. if everything is formulated in the language of second-order arithmetic. In this paper, we identify basic theorems of third-order arithmetic, e.g. concerning semi-continuous functions, such that the sequential versions have very different logical properties. In particular, depending on the constructive status of the original theorem, very different and independent choice principles are needed. Despite these differences, the associated Reverse Mathematics, working in Kohlenbach’s higher-order framework, is rather elegant and is still based at the core on weak König’s lemma.