Thus if P isthe set of all sets, we can apparently form the set Q = {Ae P| A ¢ A}, leading tothe contradictory Oe Q iff O¢€ Q. This is Russell’s paradox (see Exercise 1A)and can be avoided (in our naive discussion) by agreeing that no aggregate shallbe a set which would be an element of itself.

Cohen (Independence of the Axiomof Choice; The Independence of the Continuum Hypothesis I, I1) completed theproof of independence for each by showing neither could be deduced from theexisting axioms (by showing the negation of each could consistently be added tothe Zermelo—Fraenkel axiom scheme). See P. J. Cohen (Set Theory and theContinuum Hypothesis) for a discussion of these results and his intuition about thecontinuum hypothesis. Another expository reference is Cohen (IndependenceResults in Set Theory).

Godel (The Consistency of the Axiom of Choice and of the Generalized Con-tinuum Hypothesis with the Axioms of Set Theory) proved in 1940 that additionof either the axiom of choice or the continuum hypothesis to existing set theoreticaxioms would not produce a contradiction.

The basis for our intuitive set theory is the Zermelo—Fraenkel set theory developedby Zermelo (Untersuchungen tiber die Grundlagen der Mengenlehre J) andstrengthened by Fraenkel (Zu den Grundlagen der Cantor—Zermeloschen Mengen-lehre). Their work rests on the researches of Cantor in the 1870’s which first putmathematics firmly on a set-theoretic base. Zermelo’s work, in particular, was adirect response to the Russell paradox.

We can use the itertools.combinations function to find all possible subsets of a chord for a given cardinality.

An Euler diagram (/ˈɔɪlər/, OY-lər) is a diagrammatic means of representing sets and their relationships. They are particularly useful for explaining complex hierarchies and overlapping definitions. They are similar to another set diagramming technique, Venn diagrams. Unlike Venn diagrams, which show all possible relations between different sets, the Euler diagram shows only relevant relationships.

Kuratowski definition of an ordered pair

Willard Van Orman Quine insisted on classical, first-order logic as the true logic, saying higher-order logic was "set theory in disguise".