Much stronger large cardinal notions arise from considering strong reflection properties. Recall that the Reflection Principle  (Section 4), which is provable in ZFC, asserts that every true sentence (i.e., every sentence that holds in \(V\)) is true in some \(V_\alpha\). A strengthening of this principle to second-order sentences yields some large cardinals. For example, \(\kappa\) is strongly inaccessible if and only if every \(\Sigma^1_1\) sentence (i.e., existential second-order sentence in the language of set theory, with one additional predicate symbol) true in any  structure of the form \((V_\kappa ,\in, A)\), where \(A\subseteq V_\kappa\), is true in some \((V_\alpha ,\in ,A\cap V_\alpha)\), with \(\alpha &lt;\kappa\). The same type of reflection, but now for \(\Pi^1_1\) sentences (i.e., universal second-order sentences), yields a much stronger large cardinal property of \(\kappa\), called weak compactness. Every weakly compact cardinal \(\kappa\) is Mahlo, and the set of Mahlo cardinals smaller than \(\kappa\) is stationary. By allowing reflection for more complex second-order, or even higher-order, sentences one obtains large cardinal notions stronger than weak compactness.