Lévy obtained his Ph.D. at the Hebrew University of Jerusalem in 1958, under the supervision of Abraham Fraenkel and Abraham Robinson. Using Cohen's method of forcing, he proved several results on the consistency of various statements contradicting the axiom of choice. For example, with J. D. Halpern he proved that the Boolean prime ideal theorem does not imply the axiom of choice. He discovered the models L[x] used in inner model theory. He also introduced the notions of Lévy hierarchy of the formulas of set theory, Levy collapse and the Feferman–Levy model.
- Lévy, Azriel (1958). "The independence of various definitions of finiteness" (PDF). Fundamenta Mathematicae. 46: 1–13.
- A. Lévy: A hierarchy of formulas in set theory, Memoirs of the American Mathematical Society, 57, 1965.
- J. D. Halpern, A. Lévy: The Boolean prime ideal theorem does not imply the axiom of choice, Axiomatic Set Theory, Symposia Pure Math., 1971, 83–134.
- A. Lévy: Basic Set Theory, Springer-Verlag, Berlin, 1979, 391 pages; reprinted by Dover Publications, 2003.
- More commonly written with an accent in English sources, e.g., A. Lévy: A hierarchy of formulas in set theory, Memoirs of the American Mathematical Society, 57, 1965.
- Baumgartner, J. E. (1980). "Review: Basic set theory by Azriel Lévy" (PDF). Bull. Amer. Math. Soc. (N.S.). 3, Part 1 (1): 772–774. doi:10.1090/S0273-0979-1980-14817-X.