1. Set theory was invented by Georg Cantor. The first attempt to consider infinite sets is attributed to Bolzano (who also introduced the term Menge). It was however Cantor who realized the significance of one-to-one functions between sets and introduced the notion of cardinality of a set. Cantor originated the theory of cardinal and ordinal numbers as well as the investigations of the topology of the real line. Much of the development in the first four sections follows Cantor’s work. The main reference to Cantor’s work is his collected works, Cantor [1932]. Another source of references to the early research in set theory is Hausdorff’s book [1914].
2. Cantor started his investigations in [1874], where he proved that the set of all real numbers is uncountable, while the set of all algebraic reals is countable. In [1878] he gave the first formulation of the celebrated continuum hypothesis.
3. The axioms for set theory (except Replacement and Regularity) are due to Zermelo [1908]. Prior to Zermelo, Frege attempted to axiomatize set theory, using the false comprehension schema; Zermelo formulated his axioms following the discovery of several paradoxes by Burali-Forti, Cantor, Russell, and Zermelo. The Replacement schema is due to Fraenkel [1922a] and Skolem (see [1970, pp. 137–152]).
4. Exercises 1.12 and 1.13: Tarski [1925a].
5. The theory of well-ordered sets was developed by Cantor, who also introduced transfinite induction. The idea of identifying an ordinal number with the set of smaller ordinals is due to Zermelo and von Neumann.