The most famous large cardinals, called measurable, were discovered by Stanisław Ulam in 1930 as a result of his solution to the Measure Problem. A (two-valued) measure, or ultrafilter, on a cardinal \(\kappa\) is a subset \(U\) of \(\mathcal{P}(\kappa)\) that has the following properties: (i) the intersection of any two elements of \(U\) is in \(U\); (ii) if \(X\in U\) and \(Y\) is a subset of \(\kappa\) such that \(X\subseteq Y\), then \(Y\in U\); and (iii) for every \(X\subseteq\kappa\), either \(X\in U\) or \(\kappa -X\in U\), but not both. A measure \(U\) is called \(\kappa\)-complete if every intersection of less than \(\kappa\) elements of \(U\) is also in \(U\). And a measure is called non-principal if there is no \(\alpha &lt;\kappa\) that belongs to all elements of \(U\). A cardinal \(\kappa\) is called measurable if there exists a measure on \(\kappa\) that is \(\kappa\)-complete and non-principal.