The chaotic volume-preserving standard map is used to illustrate the invert
ibility paradox, which is related to the reversibility paradox of the micro
scopic foundations of thermodynamics. The new paradox, whose resolution rel
ies exclusively on phase-space arguments, gives insight into Boltzmann's or
iginal resolution of the reversibility paradox. (C) 1999 American Associati
on of Physics Teachers.