 In addition, Hilbert had noticed in unpublished work (see Kahle and Peckhaus 2002) that additional contradictions of a mathematical nature can arise. The first one derives from assuming that there is a well-defined set \(C\) which satisfies the following closure conditions: (i) the set \(N\) of natural numbers is an element of \(C\); (ii) \(X^X \in C\), whenever \(X \in C\) (where \(X^X\) is the set of all functions from \(X\) to \(X)\); (iii) \(\cup X \in C\), whenever \(X \subseteq C\). Then by (iii), \(\cup C = U \in C\) and finally \(F = U^U \in C\). But by definition of union, \(F \subseteq U\); hence there would exist a map of \(U\) onto \(F\) and a contradiction could be derived by diagonalization.