1979/01/01 by Andreas Blass · 1 citation
Mathematics · Psychology · Computer Science · #Advanced Topology and Set Theory #Philosophy and Theoretical Science #Advanced Algebra and Logic #Mathematics #Axiom of choice #Abelian group #Injective function #Zermelo–Fraenkel set theory #Axiom independence #Axiom #Constructive set theory #Urelement #Divisible group #Pure mathematics #Abelian category #Rank of an abelian group #Elementary abelian group #Discrete mathematics #Set (abstract data type) #Set theory
paper · pdf · doi:10.1090/s0002-9947-1979-0542870-6
openalex publication_date 1979/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/05/21
We study the connection between the axiom of choice and the principles of existence of enough projective and injective abelian groups. We also introduce a weak choice principle that says, roughly, that the axiom of choice is violated in only a set of different ways. This principle holds in all ordinary Fraenkel-Mostowski-Specker and Cohen models where choice fails, and it implies, among other things, that there are enough injective abelian groups. However, we construct an inner model of an Easton extension with no nontrivial injective abelian groups. In the presence of our weak choice principle, the existence of enough projective sets is as strong as the full axiom of choice, and the existence of enough free projective abelian groups is nearly as strong. We also prove that the axiom of choice is equivalent to “all free abelian groups are projective” and to “all divisible abelian groups are injective."