Choice, extension, conservation. From transfinite to finite proof methods in abstract algebra

Maximality principles such as the ones going back to Kuratowski and Zorn ensure the existence of higher type ideal objects the use of which is commonly held indispensable for mathematical practice. The modern turn towards computational methods, which can be witnessed to have a strong influence on contemporary foundational studies, encourages a reassessment within a constructive framework of the methodological intricacies that go along with invocations of maximality principles. The common thread that can be followed through the chapters of this thesis is explained by the attempt to put the widespread use of ideal objects under constructive scrutiny. It thus walks the tracks of a revised Hilbert’s programme which has inspired a reapproach to constructive algebra by finitary means, and for which Scott’s entailment relations have already shown to provide a vital and utmost versatile tool. In this thesis several forms of the Kuratowski-Zorn Lemma are introduced and proved equivalent over constructive set theory; the notion of Jacobson radical is brought from commutative rings to a general ideal theory for single-conclusion entailment relations; a flexible conservation criterion of Scott for multi-conclusion entailment relations is put into action; elementary and constructive variants for algebraic extension theorems such as Sikorski’s on the injectivity of complete atomic Boolean algebras are phrased and proved in terms of entailment relations; and a point-free version of Sikora’s theorem on spaces of orderings of groups is obtained by a revisitation with syntactical means of some of the classical criteria for groups to be orderable.