On the Leibniz - Mycielski Axiom in Set Theory ∗
نویسنده
چکیده
Motivated by Leibniz’s thesis on the identity of indiscernibles, Mycielski introduced a set theoretic axiom, here dubbed the Leibniz-Mycielski axiom LM, which asserts that for each pair of distinct sets x and y there exists an ordinal α exceeding the ranks of x and y, and a formula φ(v), such that (V α ,∈) satisfies φ(x) ∧ ¬φ(y) . We examine the relationship between LM and some other axioms of set theory in this paper. Our principal results, all proved assuming Con(ZF),
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Leibnizian models of set theory
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 2 א 0 nonisomorphic countable Leibnizian models. Theorem B. ...
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