Spaces

. We give a complete classification of mixed Tsirelson spaces T [( F i ,θ i ) ri =1 ] for finitely many pairs of given compact and hereditary families F i of finite sets of integers and 0 < θ i < 1 in terms of the Cantor-Bendixson indexes of the families F i , and θ i (1 ≤ i ≤ r ). We prove that there are unique countable ordinal α and 0 < θ < 1 such that every block sequence of T [( F i ,θ i ) ri =1 ] has a subsequence equivalent to a subsequence of the natural basis of the T ( S ω α ,θ ). Finally, we give a complete criterion of comparison in between two of these mixed Tsirelson spaces.