Morse–Kelley set theory
With the exception of Class Comprehension, the following axioms are the same as those for NBG, inessential details aside. The symbolic versions of the axioms employ the following notational devices:
A set and a class having the same extension are identical. Hence MK is not a two-sorted theory, appearances to the contrary notwithstanding.
The axioms and definitions in this section are, but for a few inessential details, taken from the Appendix to Kelley (1955). The explanatory remarks below are not his. The Appendix states 181 theorems and definitions, and warrants careful reading as an abbreviated exposition of axiomatic set theory by a working mathematician of the first rank. Kelley introduced his axioms gradually, as needed to develop the topics listed after each instance of Develop below.
Notations appearing below and now well-known are not defined. Peculiarities of Kelley's notation include: