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We introduce the notion of weakly extendible cardinals and show that thesecardinals are characterized in terms of weak compactness of second order logic.The consistency strength and largeness of weakly extendible cardinals arelocated strictly between that of strongly unfoldable (i.e. shrewd) cardinals,and strongly uplifting cardinals. Weak compactness of many other logics can beconnected to certain variants of the notion of weakly extendible cardinals. Wealso show that, under V=L, a cardinal $\kappa$ is the weak compactness numberof ${\cal L}^{\aleph_0,II}_{stat,\kappa,\omega}$ if and only if it is the weakcompactness number of ${\cal L}^{II}_{\kappa,\omega}$. The latter condition isequivalent to the condition that $\kappa$ is weakly extendible by thecharacterization mentioned above (this equivalence holds without the assumptionof V=L).

Weakly extendible cardinals and compactness of extended logics