Ramsey's Theorem and Reverse Mathematics

Reverse Mathematics seeks to find the minimal set existence or comprehension axioms needed to prove a theorem in second order arithmetic. Just like $ZF$ is used as a base theory to compare choice principles, a theory called $\textbf{RCA}_0$, which can formalize a part of ordinary mathematics, is used to compare theorems, axioms, and combinatorial principles by their strength. In this paper, we introduce the subfield of Reverse Mathematics by comparing Ramsey's theorem to the Big Five subsystems of second order arithmetic. We present the main subsystems of second order arithmetic and the basics of coding and recursion theory. We then show some of the basic techniques for constructing models of second order arithmetic, how they are used for independence proofs, and other techniques used in Reverse Mathematics