We examine some recently introduced extensions of Zermelo-Fraenkel set theory which, intuitively speaking, either assert Leibniz's principle on the identity of indiscernibles, or strongly deny it by positing the existence of a proper class of indiscernibles. As we shall see, Leibniz's thesis has close connections with the axiom of choice, and its antithesis is closely related to the existence of certain large cardinals (Mahlo cardinals of finite order).

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