The Morse-Kelley set theory or Morse-Kelley class theory ($MK$) 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 $NBG$. The principal difference from $NBG$ is that $MK$ allows arbitrary formulas $\phi$ 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.

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.
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