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On Strong Forms of Reflection in Set Theory

In Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo (eds.), The Hyperuniverse Project and Maximality. Basel, Switzerland: Birkhäuser. pp. 125-134 (2018)

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  1. (1 other version)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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  • (1 other version)Multiverse Conceptions in Set Theory.Carolin Antos, Sy-David Friedman, Radek Honzik & Claudio Ternullo - 2015 - Synthese 192 (8):2463-2488.
    We review different conceptions of the set-theoretic multiverse and evaluate their features and strengths. In Sect. 1, we set the stage by briefly discussing the opposition between the ‘universe view’ and the ‘multiverse view’. Furthermore, we propose to classify multiverse conceptions in terms of their adherence to some form of mathematical realism. In Sect. 2, we use this classification to review four major conceptions. Finally, in Sect. 3, we focus on the distinction between actualism and potentialism with regard to the (...)
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  • Maximality Principles in Set Theory.Luca Incurvati - 2017 - Philosophia Mathematica 25 (2):159-193.
    In set theory, a maximality principle is a principle that asserts some maximality property of the universe of sets or some part thereof. Set theorists have formulated a variety of maximality principles in order to settle statements left undecided by current standard set theory. In addition, philosophers of mathematics have explored maximality principles whilst attempting to prove categoricity theorems for set theory or providing criteria for selecting foundational theories. This article reviews recent work concerned with the formulation, investigation and justification (...)
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