A central issue in dimension reduction is choosing a sensible number of dimensions to be retained. This work demonstrates the asymptotic consistency of the maximum likelihood criterion for determining the intrinsic dimension of a dataset in a isotropic version of Probabilistic Principal Component Analysis (PPCA). Numerical experiments on simulated and real datasets show that the maximum likelihood criterion can actually be used in practice and outperforms existing intrinsic dimension selection criteria in various situations. This paper exhibits as well the limits of the maximum likelihood criterion and recommends in specific situations the use of the AIC criterion.