On the Leibniz–Mycielski axiom in set theory

Volume 181 / 2004

Ali Enayat Fundamenta Mathematicae 181 (2004), 215-231 MSC: 03E25, 03E35, 03C62. DOI: 10.4064/fm181-3-2


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] ${\rm Con(ZF)} \Rightarrow{\rm Con(ZF+LM+\lnot AC)}$.


  • Ali EnayatDepartment of Mathematics and Statistics
    American University
    4400 Massachusetts Ave. NW
    Washington, DC 20016-8050, U.S.A.

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