Zermelo Set Theory

The version of set theory obtained if Axiom 6 of Zermelo-Fraenkel set theory is replaced by

6'. Selection axiom (or "axiom of subsets"): for any set-theoretic formula , ,

which can be deduced from Axiom 6. However, there seems to be some disagreement in the literature about just which axioms of Zermelo-Fraenkel set theory constitute "Zermelo Set Theory." Mendelson (1997) does not include the axioms of choice, foundation, replacement In Zermelo set theory, but does includes 6'. However, Enderton (1977) includes the axioms of choice and foundation, but does not include the axioms of replacement or Selection.

REFERENCES:

Enderton, H. B. Elements of Set Theory. New York: Academic Press, 1977.

Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, 1997.

Iyanaga, S. and Kawada, Y. (Eds.). "Zermelo-Fraenkel Set Theory." §35B in Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 135, 1980.

Zermelo, E. "Über Grenzzahlen und Mengenbereiche." Fund. Math. 16, 29-47, 1930.