On the Leibniz–Mycielski axiom in set theory
Volume 181 / 2004
Fundamenta Mathematicae 181 (2004), 215-231
MSC: 03E25, 03E35, 03C62.
DOI: 10.4064/fm181-3-2
Abstract
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 $\alpha$ exceeding the ranks of $x$ and $y$, and a formula $\varphi(v),$ such that $(V_{\alpha},\in)$ satisfies $\varphi (x)\wedge\lnot\varphi(y)$.
We examine the relationship between LM and some other axioms of set theory. Our principal results are as follows:
1. In the presence of ZF, the following are equivalent:(a) LM.
(b) The existence of a parameter free definable class function $\bf F$ such that for all sets $x$ with at least two elements, $\emptyset\neq{\bf F}(x)\subsetneq x.$(c) The existence of a parameter free definable injection of the universe into the class of subsets of ordinals.
2. ${\rm Con(ZF)} \Rightarrow {\rm Con(ZFC+\lnot LM)}$.
3. [Solovay] ${\rm Con(ZF)} \Rightarrow{\rm Con(ZF+LM+\lnot AC)}$.
