Identification involves obtaining a model from an a priori chosen model class(es) using finite corrupted data, The corruption may be due to several reasons ranging from noise to unmodeled dynamics, since the real system may not belong to the model class, Two popular approaches - probabilistic and set-membership identification - deal with this problem by imposing temporal constraints on the noise sample paths, We differentiate between the two sources of error by imposing different types of constraints on the corruption, if the source of corruption is noise, we model it by imposing temporal constraints on the possible realizations of noise. On the other hand, if it results from unmodeled dynamics informational constraints are imposed. Contrary to probabilistic identification where the parameters of the identified model converge to the true parameters in the presence of noise, current results in set-membership identification do not have this convergence property, Our approach leads to bridging this gap between probabilistic and set-membership identification when the source! of corruption is noise. For the case when both unmodeled dynamics and noise are present, we derive consistency results far the case when the unmodeled dynamics can be described either by an linear time-invariant system or by a static nonlinearity.