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  1. Maximality Principles in the Hyperuniverse Programme.Sy-David Friedman & Claudio Ternullo - 2020 - Foundations of Science 28 (1):287-305.
    In recent years, one of the main thrusts of set-theoretic research has been the investigation of maximality principles for V, the universe of sets. The Hyperuniverse Programme (HP) has formulated several maximality principles, which express the maximality of V both in height and width. The paper provides an overview of the principles which have been investigated so far in the programme, as well as of the logical and model-theoretic tools which are needed to formulate them mathematically, and also briefly shows (...)
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  • Are Large Cardinal Axioms Restrictive?Neil Barton - 2023 - Philosophia Mathematica 31 (3):372-407.
    The independence phenomenon in set theory, while pervasive, can be partially addressed through the use of large cardinal axioms. A commonly assumed idea is that large cardinal axioms are species of maximality principles. In this paper I question this claim. I show that there is a kind of maximality (namely absoluteness) on which large cardinal axioms come out as restrictive relative to a formal notion of restrictiveness. Within this framework, I argue that large cardinal axioms can still play many of (...)
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  • Universism and extensions of V.Carolin Antos, Neil Barton & Sy-David Friedman - 2021 - Review of Symbolic Logic 14 (1):112-154.
    A central area of current philosophical debate in the foundations of mathematics concerns whether or not there is a single, maximal, universe of set theory. Universists maintain that there is such a universe, while Multiversists argue that there are many universes, no one of which is ontologically privileged. Often model-theoretic constructions that add sets to models are cited as evidence in favour of the latter. This paper informs this debate by developing a way for a Universist to interpret talk that (...)
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