Leibnizian models of set theory

Abstract

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 2^ℵ₀ 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 2^ℵ₁ 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 2^ℵ₁ nonisomorphic ℵ₁-like Leibnizian models.