Does anything require Morse-Kelley (MK) set theory as foundation?

Question

Is there any area of "non-foundations" math (areas other than things like logic and set theory) which can be formalized in MK but not ZFC (or NBG)?

I skimmed pertinent sections of Kelley’s book General Topology in which MK is axiomatized (in an appendix), and he doesn't seem to really say why he's using MK instead of ZFC (maybe just historical reasons, in that ZFC wasn't yet the de facto standard?). On the other hand I've seen statements online (can't seem to re-find them) vaguely claiming that ZFC is insufficient for some large topological spaces. Not sure what the real truth is.

Update: Edited to correctly attribute General Topology to Kelley (not Morse)

Update 2: This paper says:

We have seen that a standard textbook uses NGB and hints at superclasses of proper classes. Advanced results cited in applications use impredicative definitions of classes, which means MK rather than NGB.

From context, it seems that the “standard textbook” in question is Hartshorne’s Algebraic Geometry, but the identity of the “advanced results” mentioned are a complete mystery. Perhaps it means “advanced results” from algebraic geometry cited in Hartshorne’s book?

For what I know $\mathsf {MK}$ belongs to the "family" $\mathsf {ZFC}$ and $\mathsf {NBG}$, while Alternative Axiomatic Set Theories "are systems of set theory differing significantly from Zermelo-Frankel set theory and its close relatives". Having said that, the impact on mathematics outside set theory is not so much in the version used, but in the Large cardinal axioms assumed. — Mauro ALLEGRANZA, Jan 15 '25 at 09:47
It's NOT Morse's book, but instead it's General Topology by John Leroy Kelley (1916-1999) (1955). The appendix was also written by Kelley, but according to the acknowledgements (p. vii), the appendix is based on "the [then] unpublished system of Morse [ Anthony Perry Morse (1911-1984) ], and I am grateful for his permission to use these; he is not responsible for inaccuracies in my writing". — Dave L. Renfro, Jan 15 '25 at 12:16
Regarding Morse, my comments at the end of this MSE answer may be of interest. Also, there might be more about $\mathsf {MK}$ over $\mathsf {ZFC}$ in the preface to Morse's book A Theory of Sets (1965; 2nd edition 1986) or in reviews of Morse's book -- Current Science (20 May 1966) AND Journal of Symbolic Logic (March 1968) AND (continued) — Dave L. Renfro, Jan 15 '25 at 12:42
Canadian Mathematical Bulletin (June 1968) AND Proceedings of the Edinburgh Mathematical Society (June 1968). — Dave L. Renfro, Jan 15 '25 at 12:42
See page 250 of paper linked in Update 2: Grothendieck gave a proof of what set theorists already knew: the definition of a universe in ZFC... Since each universe models ZFC, the existence of a universe or of an uncountable strongly inaccessible cardinal is not provable in ZFC. Grothendieck’s own axiom of universes posited that every set is contained in some universe... We write ZFC+U for the more modest theory of one universe. That is, ZFC+U consists of ZFC plus the assumption of a universe, or equivalently the assumption of one uncountable strongly inaccessible cardinal. — Mauro ALLEGRANZA, Jan 16 '25 at 09:47
And see page 369: 5.1. Comparing extensions of ZFC. Both NGB and MK extend ZFC by positing a class U containing all sets. [...] The elements of proper classes are sets, and no proper class is an element of any collection in NGB or MK. The great difference is that NGB only allows quantification over sets in defining a class while MK can also quantify over classes to define a class. — Mauro ALLEGRANZA, Jan 16 '25 at 09:48
Regarding "foundations", see Penelope Maddy, Set-theoretic foundations (Cont.Math, 2017) — Mauro ALLEGRANZA, Jan 16 '25 at 10:08

Does anything require Morse-Kelley (MK) set theory as foundation?

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