()–() fulfilled), If then Thus By property (), and since , by (), () and (), It follows that The conclusion is that is an Ulam number. There is a similar proof that the supremum of a set of Ulam numbers with an Ulam number is again a Ulam number. Together with the previous result, this implies that a cardinal that is not an Ulam number is weakly inaccessible. See also Normal measure Mitchell order List of large cardinal properties Notes Citations References . . . . . . . A copy of parts I and II of this article with corrections is available at the author's web page. . . Large cardinals Determinacy