Reverse Mathematics: classifying theorems

This lecture by Keita Yokoyama provides an introduction to reverse mathematics, a field that classifies mathematical theorems based on the axioms needed to prove them. The talk begins with the historical context, tracing the idea back to Hilbert’s program and its failure due to Gödel’s incompleteness theorems. Yokoyama then explains the framework of second-order arithmetic, defining the language, formula classes, and key axioms such as induction, comprehension, and weak König’s lemma. He introduces the ‘big five’ subsystems: RCA0, WKL0, ACA0, ATR0, and Pi1-CA0, and describes their intuitive strengths and typical mathematical theorems they can prove. Examples include the intermediate value theorem (RCA0), the Brouwer fixed point theorem (WKL0), and the monotone convergence theorem (ACA0). The talk also discusses exceptions to the big five classification, such as the Baire category theorem and Ramsey’s theorem, which do not fit neatly into the hierarchy. Finally, Yokoyama illustrates the differences in logical strength using various fixed point theorems, from Banach’s contraction mapping theorem (provable in RCA0) to the Kist and K fixed point theorem (stronger, related to ATR0). The lecture concludes with a mention of ongoing research and open questions in the field.