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  1. Cantor, Choice, and Paradox.Nicholas DiBella - forthcoming - The Philosophical Review.
    I propose a revision of Cantor’s account of set size that understands comparisons of set size fundamentally in terms of surjections rather than injections. This revised account is equivalent to Cantor's account if the Axiom of Choice is true, but its consequences differ from those of Cantor’s if the Axiom of Choice is false. I argue that the revised account is an intuitive generalization of Cantor’s account, blocks paradoxes—most notably, that a set can be partitioned into a set that is (...)
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  • Partition Principles and Infinite Sums of Cardinal Numbers.Masasi Higasikawa - 1995 - Notre Dame Journal of Formal Logic 36 (3):425-434.
    The Axiom of Choice implies the Partition Principle and the existence, uniqueness, and monotonicity of (possibly infinite) sums of cardinal numbers. We establish several deductive relations among those principles and their variants: the monotonicity follows from the existence plus uniqueness; the uniqueness implies the Partition Principle; the Weak Partition Principle is strictly stronger than the Well-Ordered Choice.
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  • Realizing realizability results with classical constructions.Asaf Karagila - 2019 - Bulletin of Symbolic Logic 25 (4):429-445.
    J. L. Krivine developed a new method based on realizability to construct models of set theory where the axiom of choice fails. We attempt to recreate his results in classical settings, i.e., symmetric extensions. We also provide a new condition for preserving well ordered, and other particular type of choice, in the general settings of symmetric extensions.
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  • From abstraction and indiscernibility to classification and types.Jean-Baptiste Joinet & Thomas Seiller - 2021 - Kagaku Tetsugaku 53 (2):65-93.
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  • No decreasing sequence of cardinals.Paul Howard & Eleftherios Tachtsis - 2016 - Archive for Mathematical Logic 55 (3-4):415-429.
    In set theory without the Axiom of Choice, we investigate the set-theoretic strength of the principle NDS which states that there is no function f on the set ω of natural numbers such that for everyn ∈ ω, f ≺ f, where for sets x and y, x ≺ y means that there is a one-to-one map g : x → y, but no one-to-one map h : y → x. It is a long standing open problem whether NDS implies (...)
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  • The Axiom of Choice and the Partition Principle from Dialectica Categories.Samuel G. Da Silva - forthcoming - Logic Journal of the IGPL.
    The method of morphisms is a well-known application of Dialectica categories to set theory. In a previous work, Valeria de Paiva and the author have asked how much of the Axiom of Choice is needed in order to carry out the referred applications of such method. In this paper, we show that, when considered in their full generality, those applications of Dialectica categories give rise to equivalents of either the Axiom of Choice or Partition Principle —which is a consequence of (...)
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