Are There Enough Injective Sets?

TL;DR

This paper investigates the existence and properties of injective sets within Constructive ZF (CZF), demonstrating limitations and conditions under which injective sets can or cannot be sufficiently abundant.

Contribution

It provides new results on the non-existence of enough injective sets in CZF without classical principles and explores the duality between injective and projective sets in an intuitionistic framework.

Findings

No two-element injective sets unless excluded middle is assumed.

Power set axiom is needed for strongly enough injective sets.

It is consistent that only singleton sets are injective in CZF.

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-Loef'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.