Well-foundedness in Realizability

Abstract

However, the implication from 1) to 2) is equivalent to the principle of Dependent Choices (as the reader can show for himself) and cannot be proved in ZF. A relation < on X is called well-founded if any of these statements holds. Intuitionistically, the 3 statements are far from equivalent (hence, in formalizing an intuitionistic notion of well-foundedness, care is needed). Notion 1) is, intuitionistically, too weak to be of any use; whereas notion 2) is far too strong (by a well-known argument, as soon as an inhabited relation < satisfies 2), classical logic is forced on us). Hence we focus on notion 3), which is also usually taken as part of an axiomatization of intuitionistic set theory. The behaviour of well-founded induction for primitive recursive well-founded relations on the natural numbers in formal arithmetic, has been studied by many people. Classical results in the area are: