Countable sets versus sets that are countable in reverse mathematics

The program Reverse Mathematics (RM for short) seeks to identify the axioms necessary prove theorems of ordinary mathematics, usually working in language second-order arithmetic L 2 . A major theme RM is therefore study structures that are countable or can be approximated by sets. Now, sets must represented sequences here, because higher-order definition ‘countable set’ involving injections/bijections N cannot directly expressed Working Kohlenbach’s RM, we investigate various central theorems, e.g. those due König, Ramsey, Bolzano, Weierstrass, and Borel, their (often original) formulation based on N. This turns out closely related logical properties uncountably R, recently developed author Dag Normann. ‘being countable’ existence an injection (Kunen) a bijection (Hrbacek–Jech). former (and not latter) choice yields ‘explosive’ i.e. relatively weak statements become much stronger when combined with discontinuous functionals, even up Π 1 - CA 0 Nonetheless, replacing ‘sequence’ seriously reduces first-order strength these whatever notion ‘set’ used. Finally, obtain ‘splittings’ lemmas König from zoo, showing latter ‘a lot more tame’ formulated