On the Leibniz-Mycielski axiom in set theory

• Autorzy: Ali Enayat
• Języki publikacji: EN

Abstrakty

EN

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).

Kategorie tematyczne

• 03E25: Axiom of choice and related propositions
• 03C62: Models of arithmetic and set theory
• 03E35: Consistency and independence results

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2004
• Tom: 181
• Numer: 3
• Strony: 215-231

Daty

wydano

2004

Twórcy

autor

Ali Enayat

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

Identyfikatory

DOI

10.4064/fm181-3-2