In MK (Kelley-Morse) class theory, if i add an axiom that any cardinal except $On$ has an inaccessible greater than it (ie. essentially a Tarski/Grothendiek universe axiom), does that compel me to admit the existence of any other large cardinals (eg. measurable cardinals)?
Broadly what are the known side-effects of an ever increasing sequence of inaccessibles in MK?
For example this answer Is the axiom of universes 'harmless'? states that universes are used to resolve Fermats Last Theorem (although they can be dispensed with).