The invention provides a process for producing polyurethane foams, the foams thus produced and the uses thereof. In the inventive process compositions comprising A) isocyanate-functional prepolymers obtainable by the reaction of A1) aliphatic, low molecular weight diisocyanates of molar mass from 140 to 278 g/mol with A2) polyalkylene oxides having an OH functionality of two or more, A3) optionally further isocyanate-reactive components not covered by A2), B) water in an amount of at least 2% by weight, based on the total weight of the composition; C) optionally heterocyclic 4-membered or 6-membered ring oligomers of low molecular weight, aliphatic diisocyanates having a molar mass of 140 to 278 g/mol; D) optionally catalysts; E) optionally salts of weak acids, the corresponding free acids of which have a pKA in water at 25° C. of ≥3.0 and ≤14.0; F) optionally surfactants; G) optionally mono- or polyhydric alcohols or polyols, and H) optionally hydrophilic polyisocyanates obtainable by reaction of H1) low molecular weight, aliphatic diisocyanates of molar mass from 140 to 278 g/mol and/or polyisocyanates preparable therefrom and having an isocyanate functionality of 2 to 6 with H2) monofunctional polyalkylene oxides of OH number from 10 to 250 and of ethylene oxide content from 50 to 100 mol %, based on the total amount of the oxyalkylene groups present, are provided, foamed and cured, wherein the isocyanate-containing components, especially components A), C) and H), have a total isocyanate content within a range from 2% to 8% by weight and a content of urethane groups of 1.0 to 3.5 mol/kg, based in each case on the total amount of the isocyanate-containing components.