 The underlying set of \(\Pi A_i\) is the cartesian product \(\Pi X_i\) of the underlying sets \(X_i\), and consists of those \(I\)-tuples whose \(i\)-th element is some element of \(X_i\). (\(I\) may even be uncountable, but in this case the nonemptiness of \(\Pi X_i\) as a consequence of the nonemptiness of the individual \(X_i\)’s is equivalent to the axiom of choice. This should be kept in mind for any constructive applications of Birkhoff’s theorem.)