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 satisfies φ(x) ∧¬ φ(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 F such that for all sets x with at least two elements, ∅ ≠ F(x) ⊊ x. (c) The existence of a parameter free definable injection of the universe into the class of subsets of ordinals. 2. Con(ZF) ⇒ Con(ZFC +¬LM). 3. [Solovay] Con(ZF) ⇒ Con(ZF + LM + ¬AC).
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm181-3-2,
author = {Ali Enayat},
title = {On the Leibniz-Mycielski axiom in set theory},
journal = {Fundamenta Mathematicae},
volume = {184},
year = {2004},
pages = {215-231},
zbl = {1051.03041},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm181-3-2}
}
Ali Enayat. On the Leibniz-Mycielski axiom in set theory. Fundamenta Mathematicae, Tome 184 (2004) pp. 215-231. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm181-3-2/