The aim of this paper is to explain for broad audience the author’s result concerning the Navier–Stokes problem (NSP) in R3 without boundaries. It is proved that the NSP is contradictory in the following sense: if one assumes that the initial data v(x,0)≢0, ∇·v(x,0)=0 and the solution to the NSP exists for all t≥0, then one proves that the solution v(x,t) to the NSP has the property v(x,0)=0. This paradox shows that the NSP is not a correct description of the fluid mechanics problem and the NSP does not have a solution. In the exceptional case, when the data are equal to zero, the solution v(x,t) to the NSP exists for all t≥0 and is equal to zero, v(x,t)≡0. Thus, one of the millennium problems is solved.