Reverse mathematics and order theoretic fixed point theorems

Abstract

The theory of countable partially ordered sets (posets) is developed within a weak subsystem of second order arithmetic. We within $$\mathsf {RCA_0}$$RCA0 give definitions of notions of the countable order theory and present some statements of countable lattices equivalent to arithmetical comprehension axiom over $$\mathsf {RCA_0}$$RCA0. Then we within $$\mathsf {RCA_0}$$RCA0 give proofs of Knaster---Tarski fixed point theorem, Tarski---Kantorovitch fixed point theorem, Bourbaki---Witt fixed point theorem, and Abian---Brown maximal fixed point theorem for countable lattices or posets. We also give Reverse Mathematics results of the fixed point theory of countable posets; Abian---Brown least fixed point theorem, Davis' converse for countable lattices, Markowski's converse for countable posets, and arithmetical comprehension axiom are pairwise equivalent over $$\mathsf {RCA_0}$$RCA0. Here the converses state that some fixed point properties characterize the completeness of the underlying spaces.