ZORNS LEMMA AND COMPLETE BOOLEAN-ALGEBRAS IN INTUITIONISTIC TYPE THEORIES

We analyze Zorn's Lemma and some of its consequences for Boolean algeb ras in a constructive setting. We show that Zorn's Lemma is persistent in the sense that, if it holds in the underlying set theory, in a pro perly stated form it continues to hold in all intuitionistic type theo ries of a certain natural kind. (Observe that the axiom of choice cann ot be persistent in this sense since it implies the law of excluded mi ddle.) We also establish the persistence of some familiar results in t he theory of (complete) Boolean algebras-notably, the proposition that every complete Boolean algebra is an absolute subretract. This (almos t) resolves a question of Banaschewski and Bhutani as to whether the S ikorski extension theorem for Boolean algebras is persistent.