A system of axiomatic set theory—Part II
Paul Bernays · Journal of Symbolic Logic · 1941
For the formulation of the remaining axioms we need the notions of a function and of a one-to-one correspondence. We define a function to be a class of pairs in which different elements always have different first members; or, in other words, a class F of pairs such that, to every element a of its domain there is a unique element b of its converse domain determined by the condition 〈a, b〉ηF. We shall call the set b so determined the value of F for a, and denote it (following the mathematical usage) by F(a). A set which represents a function—i.e., a set of pairs in which different elements always have different first members—will be called a functional set. If b is the value of the function F for a, we shall say that F assigns the set b to the set a; and if a functional set f represents F, we shall say also that f assigns the set b to the set a. A class of pairs will be called a one-to-one correspondence if both it and its converse class are functions. We shall say that there exists a one-to-one correspondence between the classes A and B (or of A to B) if A and B are domain and converse domain of a one-to-one correspondence. Likewise we shall say that there exists a one-to-one correspondence between the sets a and b (or of a to b) if a and b respectively represent the domain and the converse domain of a one-to-one correspondence. In the same fashion we speak of a one-to-one correspondence between a class and a set, or a set and a class.