In the late nineteenth century, Georg Cantor created mathematical theories, first of sets or aggregates of real numbers (or linear points), and later of sets or aggregates of arbitrary elements. The relationship of element a to set A is written a∈A; it is to be distinguished from the relationship of subset B to set A, which holds if every element of B is also an element of A, and which is written B⊆A. Cantor is most famous for his theory of transfinite cardinals, or numbers of elements in infinite sets. A subset of an infinite set may have the same number of elements as the set itself, and Cantor proved that the sets of natural and rational numbers have the same number of elements, which he called ℵ0; also that the sets of real and complex numbers have the same number of elements, which he called c. Cantor proved ℵ0 to be less than c. He conjectured that no set has a number of elements strictly between these two.
In the early twentieth century, in response to criticism of set theory, Ernst Zermelo undertook its axiomatization; and, with amendments by Abraham Fraenkel, his have been the accepted axioms ever since. These axioms help distinguish the notion of a set, which is too basic to admit of informative definition, from other notions of a one made up of many that have been considered in logic and philosophy. Properties having exactly the same particulars as instances need not be identical, whereas sets having exactly the same elements are identical by the axiom of extensionality. Hence for any condition Φ there is at most one set {x|Φ(x)} whose elements are all and only those x such that Φ(x) holds, and {x|Φ(x)}={x|Ψ(x)} if and only if conditions Φ and Ψ hold of exactly the same x. It cannot consistently be assumed that {x|Φ(x)} exists for every condition Φ. Inversely, the existence of a set is not assumed to depend on the possibility of defining it by some condition Φ as {x|Φ(x)}.
One set x0 may be an element of another set x1 which is an element of x2 and so on, x0∈x1∈x2∈…, but the reverse situation, …∈y2∈y1∈y0, may not occur, by the axiom of foundation. It follows that no set is an element of itself and that there can be no universal set y={x|x=x}. Whereas a part of a part of a whole is a part of that whole, an element of an element of a set need not be an element of that set.
Modern mathematics has been greatly influenced by set theory, and philosophies rejecting the latter must therefore reject much of the former. Many set-theoretic notations and terminologies are encountered even outside mathematics, as in parts of philosophy:
pair {a,b} {x|x=a or x=b}
singleton {a} {x|x=a}
empty set ∅ {x|x≠x}
union ∪X {a|a∈A for some A∈X}
binary union A∪B {a|a∈A or a∈B}
intersection ∩X {a|a∈A for all A∈X}
binary intersection A∩B {a|a∈A and a∈B}
difference A−B {a|a∈A and not a∈B}
complement A−B
power set ℘(A) {B|B⊆A}
(In contexts where only subsets of A are being considered, A-B may be written -B and called the complement of B.)
While the accepted axioms suffice as a basis for the development not only of set theory itself, but of modern mathematics generally, they leave some questions about transfinite cardinals unanswered. The status of such questions remains a topic of logical research and philosophical controversy.