A Constructive Picture of Noetherian Conditions and Well Quasi-orders

Gabriele Buriola, Peter Schuster, Ingo Blechschmidt

Abstract: From a constructive perspective the many notions of Noetherianity and well quasi-order form a rich landscape, which we here explore. Besides the well-studied conditions about sequences, we include the finite basis property of the original Higman lemma, trying a first joint analysis of Noetherianity and well quasi-order in the spirit of reverse mathematics with intuitionistic logic. Applying a topological semantics for intuitionistic logic, we settle a conjecture by Ray Mines; moreover, by the realizability topos of infinite-time Turing machines, we separate the ascending chain condition with finite generation from the one without.