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Studia Logica

, Volume 101, Issue 3, pp 467–482 | Cite as

Are There Enough Injective Sets?

  • Peter Aczel
  • Benno van den Berg
  • Johan Granström
  • Peter SchusterEmail author
Article
  • 149 Downloads

Abstract

The axiom of choice ensures precisely that, in ZFC, every set is projective: that is, a projective object in the category of sets. In constructive ZF (CZF) the existence of enough projective sets has been discussed as an additional axiom taken from the interpretation of CZF in Martin-Löf’s intuitionistic type theory. On the other hand, every non-empty set is injective in classical ZF, which argument fails to work in CZF. The aim of this paper is to shed some light on the problem whether there are (enough) injective sets in CZF. We show that no two element set is injective unless the law of excluded middle is admitted for negated formulas, and that the axiom of power set is required for proving that “there are strongly enough injective sets”. The latter notion is abstracted from the singleton embedding into the power set, which ensures enough injectives both in every topos and in IZF. We further show that it is consistent with CZF to assume that the only injective sets are the singletons. In particular, assuming the consistency of CZF one cannot prove in CZF that there are enough injective sets. As a complement we revisit the duality between injective and projective sets from the point of view of intuitionistic type theory.

Keywords

Injective object Constructive set theory Axiom of powerset Intuitionistic type theory Axiom of choice 

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Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Peter Aczel
    • 1
  • Benno van den Berg
    • 2
  • Johan Granström
    • 3
    • 4
  • Peter Schuster
    • 5
    Email author
  1. 1.School of Mathematics and Computer ScienceUniversity of ManchesterManchesterUK
  2. 2.Mathematisch InstituutUniversiteit UtrechtUtrechtThe Netherlands
  3. 3.Mathematisches InstitutUniversität MünchenMünchenGermany
  4. 4.Google ZürichZürichSwitzerland
  5. 5.Department of Pure MathematicsUniversity of LeedsLeedsUK

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