The Morse-Kelley set theory or Morse-Kelley class theory () 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 . The principal difference from is that allows arbitrary formulas 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
For discussion about the category of classes in Morse-Kelly class theory, see
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