A model is said to be Leibnizian if it has no pair of indiscernibles. Mycielski has shown that there is a first order axiom LM (the Leibniz-Mycielski axiom) such that for any completion T of Zermelo-Fraenkel set theory ZF, T has a Leibnizian model if and only if T proves LM. Here we prove: Theorem 1 Every complete theory T extending ZF+LM has 2ℵ0 nonisomorphic countable Leibnizian models. Theorem 2 If κ is a prescribed definable infinite cardinal of a complete theory T extending ZF+V=OD, then there are 2ℵ1 nonisomorphic Leibnizian models 𝔐 of T of power ℵ1 such that (κ+)𝔐 is ℵ1-like. Theorem 3 Every complete theory T extending ZF+V=OD has 2ℵ1 nonisomorphic ℵ1-like Leibnizian models.
"Leibnizian models of set theory." J. Symbolic Logic 69 (3) 775 - 789, September 2004. https://doi.org/10.2178/jsl/1096901766