The strength of the minimum principle in constructive reverse mathematics

Abstract

The minimum principle states that every real-valued uniformly continuous function f : X -> R on a compact metric space X attains its minimum. Working in the base system EL0 of constructive reverse mathematics, we show that the minimum principle is equivalent to weak Konig's lemma and to the Cantor intersection theorem (i.e., the contrapositive of the Heine-Borel theorem). Moreover, the above equivalence naturally extends to the equivalence between the corresponding principles with unique existence assumptions, which are then shown to be equivalent to the Fan theorem for detachable bars and the positivity property for a real-valued uniformly continuous function on a compact space. These results sharpen the equivalence among these principles obtained by Ishihara [Math. Log. Q. (1990), 237-240] and Berger and Ishihara [Math. Log. Q. (2005), 360-364] in Bishop's constructive mathematics.