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

Lifting proof theory to the countable ordinals II: second-order indescribable cardinals