one-to-one correspondence

Two sets can be put into one-to-one correspondence when to each element of one there corresponds one element of the other, and to each distinct element of the one a different element of the other. Counting is an operation that puts n-membered sets of objects into one-to-one correspondence with the set of the first n natural numbers. When two sets can be put into such a correspondence they are equinumerous.

This definition makes no use of numbers, and opens the way to defining a number in terms of a set of equinumerous sets. It thus lies at the heart of the logicist programme of Frege and Russell. See also Hume's principle.