Universality in Set Theories: A Study in Formal Ontology (Philosophische Analyse / Philosophical Analysis, 36)

The book discusses the fate of universality and a universal set in several set theories. The book aims at a philosophical study of ontological and conceptual questions around set theory. Set theories are ontologies. They posit sets and claim that these exhibit the essential properties laid down in the set theoretical axioms. Collecting these postulated entities quantified over poses the problem of universality. Is the collection of the set theoretical entities itself a set theoretical entity? What does it mean if it is, and what does it mean if it is not? To answer these questions involves developing a theory of the universal set. We have to ask: Are there different aspects to universality in set theory, which stand in conflict to each other? May inconsistency be the price to pay to circumvent ineffability? And most importantly: How far can axiomatic ontology take us out of the problems around universality? The book discusses the fate of universality and a universal set in several set theories. The book aims at a philosophical study of ontological and conceptual questions around set theory. Set theories are ontologies. They posit sets and claim that these exhibit the essential properties laid down in the set theoretical axioms. Collecting these postulated entities quantified over poses the problem of universality. Is the collection of the set theoretical entities itself a set theoretical entity? What does it mean if it is, and what does it mean if it is not? To answer these questions involves developing a theory of the universal set. We have to Are there different aspects to universality in set theory, which stand in conflict to each other? May inconsistency be the price to pay to circumvent ineffability? And most How far can axiomatic ontology take us out of the problems around universality? Die Reihe bietet ein Publikationsforum fr innovative Arbeiten zu allen Themengebieten der analytischen Philosophie. Die Bnde in dieser Reihe erscheinen in deutscher oder englischer Sprache. INTRODUCTION I - ITERATION, FOUNDATION, AND REFLECTION II - LIMITATIONS OF SIZE III - VIRTUAL SETS AND CONSTRUCTIVISM IV - CONSISTENT SET THEORIES WITH UNIVERSAL SET V - PARACONSISTENT SET THEORIES WITH A UNIVERSAL SET VI - BROADER ONTOLOGICAL REFLECTIONS ABBREVIATIONS AND NOTATION REFERENCES Discusses the fate of universality and a universal set in several set theories. This book provides a philosophical study of ontological and conceptual questions around set theory.