  We show that the existence of a Pi^{1}_{N}-indescribable cardinal over the
Zermelo-Fraenkel's set theory ZF is proof-theoretically reducible to iterations
of Mostowski collapsings and lower Mahlo operations. Furthermore we describe a
proof-theoretic bound on definable countable ordinals whose existence is
provable from the existence of second order indescribable cardinals over ZF.
