 The task Kolmogorov set himself in the paper is to explain why “the illegitimate use of the principle of the excluded middle” as revealed in Brouwer’s writings “has not yet led to contradictions and also why the very illegitimacy has often gone unnoticed” (van Heijenoort 1967: 416). In effect, as other passages make clear (van Heijenoort 1967: 429–430), the (unachieved) aim is to show that classical mathematics is translatable into intuitionistic mathematics, and thereby give a consistency proof of classical mathematics relative to intuitionistic mathematics.