 Two sets are said to be of the same cardinality just in case there is a bijection between them. This is standard. But we then proceed to define \(|A|\) (the cardinality of a set \(A)\) as the set of all sets which are the same size as \(A\), realizing the definition intended by Frege and Russell, and apparently intended by Cantor as well. Notice that \(|A|\) is one type higher than \(A\). The Frege natural numbers are the same objects as the finite cardinal numbers.