Does Choice Really Imply Excluded Middle? Part II: Historical,Philosophical,and Foundational Reflections on the Goodman-Myhill Result

摘要

Our regimentation of Goodman and Myhill's proof of Excluded Middle revealed among its premises a form of Choice and an instance of Separation. Here we revisit Zermelo's requirement that the separating property be definite.The instance that Goodman and Myhill used is not constructively warranted. It is that principle,and not Choice alone,that precipitates Excluded Middle. Separation in various axiomatizations of constructive set theory is examined. We conclude that insufficient critical attention has been paid to how those forms of Separation fail,in light of the Goodman-Myhill result,to capture a genuinely constructive notion of set.