nLab Morse-Kelley set theory



The Morse-Kelley set theory or Morse-Kelley class theory (MKMK) is an axiomatic approach to class theory and set theory which has both classes and sets. Whereas NBG (which also has both classes and sets) is conservative over ZFC, Morse–Kelley is not a conservative extension of NBGNBG. The principal difference from NBGNBG is that MKMK allows arbitrary formulas ϕ\phi appearing in the class comprehension axiom schema (in particular, formulae with quantifiers ranging over classes themselves).

The approach is explained in the appendix to John Kelley‘s 1955 book General Topology.

See also


A definitive source (by one of the authors of the theory) is

  • Anthony P. Morse, A theory of sets, Pure and Applied Mathematics XVIII, Academic Press (1965), xxxi+130 pp. Second Edition, Pure and Applied Mathematics 108, Academic Press (1986), xxxii+179 pp. ISBN: 0-12-507952-4

For discussion about the category of classes in Morse-Kelly class theory, see

  • Henrik Forssell, Categorical Models of Intuitionistic Theories of Sets and Classes. Master’s thesis, Carnegie Mellon University, 2004. (PDF)

Last revised on November 19, 2022 at 14:58:39. See the history of this page for a list of all contributions to it.