 Although the \(\lambda\)-calculus is ‘about’ calculating with functions by substituting values for arguments, this simple point of view cannot support a semantics for the (untyped) \(\lambda\)-calculus if by ‘function’ we understand, as is standard in set theory, a relation \(R\) such that for every pair \((x,y)\) and \((x,z)\) in \(R\) with the same first component \(x\) we have \(y = z\). For sets \(X\) and \(Y\), let \(X^Y\) denote the set of functions whose domain is \(Y\) and with values in \(X\). Intuitively, if \(X\) is the domain of an interpretation of \(\lambda\)-calculus, then \(X\) should be, in some sense, isomorphic to \(X^X\) because the domain should be closed under abstraction (as well as application). Taken literally, though, this isomorphism is impossible, because the cardinality of \(X\) always is of strictly smaller than the cardinality of \(X^X\).