On the Leibniz-Mycielski axiom in set theory

Ali Enayat. "On the Leibniz-Mycielski axiom in set theory." Fundamenta Mathematicae 181.3 (2004): 215-231. <http://eudml.org/doc/282624>.

@article{AliEnayat2004,
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 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. 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).},
author = {Ali Enayat},
journal = {Fundamenta Mathematicae},
keywords = {identity of indiscernibles; Leibniz-Mycielski axiom; axiom of choice},
language = {eng},
number = {3},
pages = {215-231},
title = {On the Leibniz-Mycielski axiom in set theory},
url = {http://eudml.org/doc/282624},
volume = {181},
year = {2004},
}

TY - JOUR
AU - Ali Enayat
TI - On the Leibniz-Mycielski axiom in set theory
JO - Fundamenta Mathematicae
PY - 2004
VL - 181
IS - 3
SP - 215
EP - 231
AB - 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. 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).
LA - eng
KW - identity of indiscernibles; Leibniz-Mycielski axiom; axiom of choice
UR - http://eudml.org/doc/282624
ER -