The weak choice principle WISC may fail in the category of sets

Abstract

The set-theoretic axiom WISC states that for every set there is a set of surjections to it cofinal in all such surjections. By constructing an unbounded topos over the category of sets and using an extension of the internal logic of a topos due to Shulman, we show that WISC is independent of the rest of the axioms of the set theory given by a well-pointed topos. This also gives an example of a topos that is not a predicative topos as defined by van den Berg.

v2 Change of title and abstract; v3 Almost completely rewritten after referee pointed out critical mistake. v4 Final version. Will be published in Studia Logica. License is CC-BY

References in corpus (3)

• Internal categories, anafunctors and localisations
• Stack semantics and the comparison of material and structural set theories
• Embedding Orders Into Cardinals With $D{C}_{\kappa }$

Cited by in corpus (6)

• Quotients, inductive types, and quotient inductive types
• W-Types with Reductions and the Small Object Argument
• The elementary construction of formal anafunctors
• Broad Infinity and Generation Principles
• Constructing Initial Algebras Using Inflationary Iteration
• An Initial Algebra Theorem Without Iteration