nLab Morse-Kelley set theory

Idea

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.

See also

• class theory

• set theory

References

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)