| Résume | We survey recent results in reverse mathematics, highlighting theorems from the general mathematics literature whose logical strength is unusual in some way. The Rival-Sands theorem for partial orders is a Ramsey-theoretic result concerning chains in partial orders of finite width. We show that the Rival-Sands theorem is equivalent to the ascending/descending sequence principle, which is a weak consequence of Ramsey's theorem for pairs (joint with Fiori Carones, Marcone, and Soldà). This gives the first example of a theorem from the modern literature that is characterized by the ascending/descending sequence principle. Ekeland's variational principle concerns approximate minima of lower semi-continuous functions that are bounded below. We show that the localized version of Ekeland's variational principle is equivalent to Pi^1_1-CA_0, even when restricted to continuous functions (joint with Fernández-Duque and Yokoyama). This is unusual because the much weaker system ACA_0 typically suffices to prove theorems about continuous functions. Caristi's fixed point theorem is a consequence of Ekeland's variational principle that concerns fixed points of arbitrary functions that are controlled by lower semi-continuous functions. We show that Caristi's theorem for Borel functions is equivalent to Towsner's transfinite leftmost path principle and therefore has the unusual position of being strictly between ATR_0 and Pi^1_1-CA_0 (joint with Fernández-Duque, Towsner, and Yokoyama). |
