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In this note, we investigate different concepts of nonlinear identifiability in the generic sense. We work in the linear algebraic framework. Necessary and sufficient conditions are found for geometrical identifiability, algebraic identifiability and identifiability with known initial conditions. Relationships between different concepts are characterized. Constructive procedures are worked out for both generic geometrical and algebraic identifiability of nonlinear systems. As an application of the theory developed, we study the identifiability properties of a four dimensional model of HIV/AIDS. The questions answered in this study include the minimal number of measurement of the variables for a complete determination of all parameters and the best period of time to make such measurements. This information will be useful in formulating guidelines for clinical practice.