nLab axiom of constructibility

Idea

The axiom of constructibility in set theory states that the constructible universe $L$ and the von Neumann universe $V$ are identical to each other.

In the absence of the axiom of replacement, one can’t actually construct the constructible universe $L$, so one has to postulate the constructible universe $L$ in the set theory via an axiom schema before formulating the axiom of constructibility.

Properties

In IZF, the axiom of constructibility does not imply the axiom of choice or excluded middle.

In ZFC and IZF, the axiom of constructibility is inconsistent with measurable cardinals.

The axiom of constructibility in ZFC is equivalent to the statement that every exact functor $F: Set \to Set$ is naturally isomorphic to the identity functor on the category of sets. However, this equivalence fails in IZF.

Related concepts

• ultimate L conjecture

References

• Wikipedia, Axiom of constructibility

• Adrian Mathias: The strength of Mac Lane set theory, Annals of Pure and Applied Logic 110 (1-3):107-234 (2001) [doi:10.1016/S0168-0072(00)00031-2]

The axiom of constructibility in constructive set theory:

• Richard Matthews, Michael Rathjen, Constructing the Constructible Universe Constructively, Annals of Pure and Applied Logic, Volume 175, Issue 3, March 2024. [doi:10.1016/j.apal.2023.103392, arxiv:2206.08283]