Definition of "set theory"

We mentioned previously that certain paradoxes in set theory arose shortly after Cantor's works were published.

1972, Peter W. Zehna, Robert Leo Johnson, Elements of Set Theory, Allyn & Bacon, page 4

The above argument, known as Russell's Paradox, was discovered by Bertrand Russell in 1901. Set theory itself began a few decades earlier with the work of George Cantor.

1984, Robert Goldblatt, Topoi, the categorial analysis of logic‎, page 9

The serious study of models of axiomatic set theories depends heavily on methods from mathematical logic which are outside the scope of these Notes.

1994, Yiannis N. Moschovakis, Notes on Set Theory, Springer, page 239

The one thing that all alternative set theories have in common is the fact that they are alternatives to ZF or ZFC.

2012, M. Randall Holmes, Thomas Forster, Thierry Libert, “Alternative Set Theories”, in Dov M. Gabbay, Akihiro Kanimori, John Woods, editors, Sets and Extensions in the Twentieth Century, North-Holland: Elsevier, page 559