have uncountable cofinality in which case this condition is redundant. Some authors add the condition that α is in X, which in practice usually makes little difference as it is often automatically satisfied. For a fixed regular uncountable cardinal κ, the Mahlo operation induces an operation on the Boolean algebra of all subsets of κ modulo the non-stationary ideal. The Mahlo operation can be iterated transfinitely as follows: M0(X) = X Mα+1(X) = M(Mα(X)) If α is a limit ordinal then Mα(X) is the intersection of Mβ(X) for β<α These iterated Mahlo operations produce the classes of α-Mahlo cardinals starting with the class of strongly inaccessible cardinals. It is also possible to diagonalize this process by defining MΔ(X) is the set of ordinals α that are in Mβ(X) for β<α. And of course this diagonalization process can be iterated too. The diagonalized Mahlo operation produces the hyper-Mahlo cardinals, and so on. Mahlo cardinals and reflection principles Axiom F is the statement that every normal function on the ordinals has a regular fixed point. (This is not a first-order axiom as it quantifies over all normal functions, so it can be considered either as a second-order axiom or as an axiom scheme.) A cardinal is called Mahlo if every normal function on it has a regular fixed point, so axiom F is in some sense saying that the class of all ordinals is Mahlo. A cardinal κ is Mahlo if and only if a second-order form of axiom F holds in Vκ. Axiom F is in turn equivalent to the statement that for any formula φ with parameters there are arbitrarily large inaccessible ordinals α such that Vα reflects φ (in other words φ holds in Vα if and only if it holds in the whole universe) . Appearance in Borel diagonalization has shown that existence of Mahlo cardinals is a necessary assumption in a sense to prove certain theorems about Borel functions on products of the closed unit interval. Let be , the -fold iterated Cartesian product of the closed unit interval with itself. The group of all permutations of that move only finitely many natural numbers can be seen as acting on by permuting coordinates. The group action also acts diagonally on any of the products , by defining an abuse of notation . For , let if and are in the same orbit under this diagonal action. Let