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 LeibnizMycielski axiom) such that for any completion T of Zermelo-Fraenkel set theory ZF , T has a Leibnizian model iff T proves LM. Here we prove: Theorem A. Every complete theory T extending ZF + LM has 20 nonisomorphic countable Leibnizian models. Theorem B. If κ is a prescribed definable infinite cardinal of a complete theory T extending ZF+ V =OD, then there are 21 nonisomorphic Leibnizian models M of T of power א1 such that (κ) M is א1-like. Theorem C. Every complete theory T extending ZF+ V = OD has 21 nonisomorphic א1-like Leibnizian models. ∗2000 Mathematics Subject Classification: 03C62, 03C50; Secondary 03H99.
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