On the Leibniz–Mycielski axiom in set theory

Ali Enayat¹

¹ Department of Mathematics and Statistics American University 4400 Massachusetts Ave. NW Washington, DC 20016-8050, U.S.A.

Fundamenta Mathematicae, Tome 181 (2004) no. 3, pp. 215-231

Résumé

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

Détail

DOI : 10.4064/fm181-3-2

Mots-clés : motivated leibnizs thesis identity indiscernibles mycielski introduced set theoretic axiom here dubbed leibniz mycielski axiom which asserts each pair distinct sets there exists ordinal alpha exceeding ranks formula varphi alpha satisfies varphi wedge lnot varphi examine relationship between other axioms set theory principal results follows presence following equivalent existence parameter definable class function sets least elements emptyset neq subsetneq existence parameter definable injection universe class subsets ordinals con rightarrow con zfc lnot solovay con rightarrow con lnot