Cantorian set theory is founded on the principles of extension and abstraction, described above. To describe some results based upon these principles, the notion of equivalence of sets will be defined. The idea is that two sets are equivalent if it is possible to pair off members of the first set with members of the second, with no leftover members on either side. To capture this idea in set-theoretic terms, the set A is defined as equivalent to the set B (symbolized by A ≡ B) if and only if there exists a third set the members of which are ordered pairs such that: (1) the first member of each pair is an element of A and the second is an element of B, and (2) each member of A occurs as a first member and each member of B occurs as a second member of exactly one pair. Thus, if A and B are finite and A ≡ B, then the third set that establishes this fact provides a pairing, or matching, of the elements of A with those of B. Conversely, if it is possible to match the elements of A with those of B, then A ≡ B, because a set of pairs meeting requirements (1) and (2) can be formed—i.e., if a ∊ A is matched with b ∊ B, then the ordered pair (a, b) is one member of the set. By thus defining equivalence of sets in terms of the notion of matching, equivalence is formulated independently of finiteness. As an illustration involving infinite sets, ℕ may be taken to denote the set of natural numbers 0, 1, 2, … (some authors exclude 0 from the natural numbers). Then {(n, n²) | n ∊ ℕ} establishes the seemingly paradoxical equivalence of ℕ and the subset of ℕ formed by the squares of the natural numbers.

As stated previously, a set B is included in, or is a subset of, a set A (symbolized by B ⊆ A) if every element of B is an element of A. So defined, a subset may possibly include all of the elements of A, so that A can be a subset of itself. Furthermore, the empty set, because it by definition has no elements that are not included in other sets, is a subset of every set.

If every element of set B is an element of set A, but the converse is false (hence B ≠ A), then B is said to be properly included in, or is a proper subset of, A (symbolized by B ⊂ A). Thus, if A = {3, 1, 0, 4, 2}, both {0, 1, 2} and {0, 1, 2, 3, 4} are subsets of A; but {0, 1, 2, 3, 4} is not a proper subset. A finite set is nonequivalent to each of its proper subsets. This is not so, however, for infinite sets, as is illustrated with the set ℕ in the earlier example. (The equivalence of ℕ and its proper subset formed by the squares of its elements was noted by Galileo Galilei in 1638, who concluded that the notions of less than, equal to, and greater than did not apply to infinite sets.)

Cardinality and transfinite numbers

The application of the notion of equivalence to infinite sets was first systematically explored by Cantor. With ℕ defined as the set of natural numbers, Cantor’s initial significant finding was that the set of all rational numbers is equivalent to ℕ but that the set of all real numbers is not equivalent to ℕ. The existence of nonequivalent infinite sets justified Cantor’s introduction of “transfinite” cardinal numbers as measures of size for such sets. Cantor defined the cardinal of an arbitrary set A as the concept that can be abstracted from A taken together with the totality of other equivalent sets. Gottlob Frege, in 1884, and Bertrand Russell, in 1902, both mathematical logicians, defined the cardinal number of a set A somewhat more explicitly, as the set of all sets that are equivalent to A. This definition thus provides a place for cardinal numbers as objects of a universe whose only members are sets.

The above definitions are consistent with the usage of natural numbers as cardinal numbers. Intuitively, a cardinal number, whether finite (i.e., a natural number) or transfinite (i.e., nonfinite), is a measure of the size of a set. Exactly how a cardinal number is defined is unimportant; what is important is that if and only if A ≡ B.

Britannica Quiz

Numbers and Mathematics

To compare cardinal numbers, an ordering relation (symbolized by <) may be introduced by means of the definition if A is equivalent to a subset of B and B is equivalent to no subset of A. Clearly, this relation is irreflexive and transitive: and imply .

When applied to natural numbers used as cardinals, the relation < (less than) coincides with the familiar ordering relation for ℕ, so that < is an extension of that relation.

The symbol ℵ₀ (aleph-null) is standard for the cardinal number of ℕ (sets of this cardinality are called denumerable), and ℵ (aleph) is sometimes used for that of the set of real numbers. Then n < ℵ₀ for each n ∊ ℕ and ℵ₀ < ℵ.

This, however, is not the end of the matter. If the power set of a set A—symbolized P(A)—is defined as the set of all subsets of A, then, as Cantor proved, for every set A—a relation that is known as Cantor’s theorem. It implies an unending hierarchy of transfinite cardinals: . Cantor proved that and suggested that there are no cardinal numbers between ℵ₀ and ℵ, a conjecture known as the continuum hypothesis.

There is an arithmetic for cardinal numbers based on natural definitions of addition, multiplication, and exponentiation (squaring, cubing, and so on), but this arithmetic deviates from that of the natural numbers when transfinite cardinals are involved. For example, ℵ₀ + ℵ₀ = ℵ₀ (because the set of integers is equivalent to ℕ), ℵ₀ · ℵ₀ = ℵ₀ (because the set of ordered pairs of natural numbers is equivalent to ℕ), and c + ℵ₀ = c for every transfinite cardinal c (because every infinite set includes a subset equivalent to ℕ).

The so-called Cantor paradox, discovered by Cantor himself in 1899, is the following. By the unrestricted principle of abstraction, the formula “x is a set” defines a set U; i.e., it is the set of all sets. Now P(U) is a set of sets and so P(U) is a subset of U. By the definition of < for cardinals, however, if A ⊆ B, then it is not the case that . Hence, by substitution,. But by Cantor’s theorem,. This is a contradiction. In 1901 Russell devised another contradiction of a less technical nature that is now known as Russell’s paradox. The formula “x is a set and (x ∉ x)” defines a set R of all sets not members of themselves. Using proof by contradiction, however, it is easily shown that (1) R ∊ R. But then by the definition of R it follows that (2) (R ∉ R). Together, (1) and (2) form a contradiction.