Choice, extension, conservation. From transfinite to finite methods in abstract algebra Daniel Wessel Universita` degli Studi di Trento Universita` degli Studi di Verona December 3, 2017 Doctoral thesis in Mathematics Joint doctoral programme in Mathematics, 30th cycle Department of Mathematics, University of Trento Department of Computer Science, University of Verona Academic year 2017/18 Supervisor: Peter Schuster, University of Verona Trento, Italy December 3, 2017 Abstract 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 com- mutative 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 entail- ment 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. Acknowledgements First and foremost, I would like to express my sincere gratitude to Peter Schuster for his guid- ance, support, and years of encouragement. Without the further help of Davide Rinaldi, who took interest in my studies upon joining our group in Verona in early autumn 2016, and who shared many an idea and insight rather generously, it would have been less likely for this thesis to take form and develop content. I thank Stefano Baratella, Emanuele Bottazzi, and Roberto Zunino for welcoming me in Trento, for stimulating discussions, and, above all, for lending an ear. As time went by, I have had the chance to meet the many inspiring visitors of the Department of Computer Science in Verona; this was a privilege through and through. Marco Benini, Tatsuji Kawai, and Samuele Maschio have never failed to cheer me up and spark my interest anew. { Thank you! Roberto Civino, Giulia Tini, and all my fellow students in Trento at some time helped to pull through. Karla Misselbeck had to put tremendous effort and a lot of patience into setting my understanding straight. Last but not least, I am very honoured that Hajime Ishihara and Stefan Neuwirth unhesitantly agreed on refereeing this thesis. Once more, a wholehearted thank you to everyone. Trento Daniel Wessel December 2017 Publications included in this thesis This thesis contains material which has already been published or has been submitted for publi- cation: • Peter Schuster and Daniel Wessel. \A general extension theorem for directed-complete partial orders". Preprint, submitted. 2017. • Peter Schuster and Daniel Wessel. \Logical completeness and Jacobson radicals". Preprint. 2017. • Davide Rinaldi, Peter Schuster, and Daniel Wessel. \Eliminating disjunctions by disjunction elimination". In: Bulletin of Symbolic Logic 23.2 (2017), pp. 181{200. • Davide Rinaldi, Peter Schuster, and Daniel Wessel. \Eliminating disjunctions by disjunction elimination". In: Indagationes Mathematicae (2017). Virtual Special Issue { L.E.J. Brouwer, fifty years later. • Davide Rinaldi and Daniel Wessel. \Extension by conservation. Sikorski's theorem". Preprint, submitted. 2016. url: https://arxiv.org/pdf/1612.07345.pdf. • Davide Rinaldi and Daniel Wessel. \Some constructive extension theorems for distributive lattices". Preprint, submitted. 2017. • Daniel Wessel. \Ordering groups syntactically". Preprint, submitted. 2017. • Peter Schuster and Daniel Wessel. \Suzumura consistency, an alternative approach". In: Journal of Applied Logic{IfCoLog (2017). To appear. Preface Maximality principles such as the ones going back to Kuratowski and Zorn stipulate the existence of higher type ideal objects without so much as accounting for an effective procedure to bear witness. This concerns a commonplace lack of computational justification which forces a strong ontological commitment, yet prevents any means of epistemic access. Throughout mathematics, typical examples arise from the alleged need for totality where partiality abounds, e.g., if maps are to be extended from sub- to ambient structures in a coherent manner, the classical solution to which may sometimes nearly be held to ridicule the intuitive and `procedural' concept of a function. If taking a rather practical stance, one is confronted with foundational issues that are inherently difficult to address. Needless to say, this describes unfounded worries from a classical point of view. But even a finitist adversary might vindicate all this quickly, presumably on the grounds of deeming those ideal objects to be mere fictions. Yet the sheer amount of deep results obtained by ideal methods, generations worth of achievements, and last but not least everyday curricula can hardly be denied, disregarded, or overruled|all of which reinforces a commonly held belief that ideal objects are indispensable for mathematical practice: be it for a far-reaching development of contemporary abstract algebra, or even to address matter-of-fact questions stemming from an economic theory of preferences. But then again, dropping the subject on the former, whether ideal methods should have found their way into the latter discipline might very well be worth an argument. I hasten to add, with all due emphasis, that this thesis does not set out to do away with misconceptions, and it does not intend to clarify conclusively whether the above rests on a mis- conception at all. Nor does it advocate a large-scale reapproach. But the overall turn towards computational methods, which today can be witnessed to have a strong influence on the founda- tions of mathematics, seems to encourage a reassessment of the methodological intricacies that go along with invocations of maximality principles in a constructive framework. It is here that some steps shall be taken. The common thread which 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. Genesis and context The studies that have led to this thesis took motivation from Bell's [33] dictum that the Kuratowski- Zorn lemma (henceforth KZL) be \constructively neutral"|as opposed to the Axiom of Choice, which has long been known to be incompatible with intuitionistic logic|along with a methodolog- ical discussion and an assessment that deems KZL to be of comparatively little use unless applied in a classical setting. It appeared reasonable that a similar analysis could be bestowed upon Raoult's principle of Open Induction, which in recent times has caused an attentional shift [36, 73, 223]. Eventually, such an analysis has not been carried out thoroughly, yet to a large extent the later development of this thesis can be traced back as to have originated in this context. It stems from the endeavour to rephrase several prominent applications of KZL (e.g., characterizations of injective objects, and orderability criteria for algebraic structures) in a manner which puts strong emphasis on constructions which sometimes are lurking in the background; and which allows to unveil their computational underpinning, if at least from a liberal, non-formal point of view. v Preface With its aims declared, this thesis walks on the tracks of a revised Hilbert's programme [243, 256] which has inspired a reapproach to constructive algebra by finitary means [94], and for which Scott's entailment relations [229] have already shown to provide a vital and utmost versatile tool ([62, 82]; see Chapter 3 for a wealth of references, and Chapter 4 for a thorough introduction). The notion of ideal element, which above has already been insinuating Hilbert's terminology, will here be understood as model of an entailment relation. While we cannot circumvent transfinite methods in order to assert the existence of such an object in general, it could well be argued that entailment relations are \couched in cognitively accessible terms" [126]. They allow for a replacement of ideal semantics, which more often than not necessitates classical reasoning, in a direct manner by formal, syntactical correspondents. For instance, rather than resorting to model existence principles, we aim at asserting consistency, and at providing an elementary proof for any claim of the latter. It is then tempting to say that we obtain constructive versions of classical theorems. Surely it is in order to quote Coquand and Lombardi [82], the work of whom has had an unmistakable influence on this thesis: \When we