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?
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