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.