Cardinality of a set V vs cardinality of its ordinal Type ord(V)

Question

I know that ordinal numbers use to position (rank) the members of a set. The order type is a relation between two sets A and B that are order isomorphic, there is a bijection between them. Sometimes the order type of a well ordered set is identified with the corresponding ordinal number. For example the order type of natural numbers is ω. Suppose V (set of even ordinals less than ω.2 + 7) is: V = {0,2,4,...; ω,ω+2,ω+4,...;ω.2,ω.2+2, ω.2 +4, ω.2+6} I read somewhere that the order type of V above is: ord(V) = ω.2+4 = {0,1,2,...;ώ,ω+1,ω+2,...;ω.2,ω.2+1,ω.2+2,ω.2+3}. My question is the following: If order number is a set that gives an order or rank to the initial set in question, how can ord(V) as defined, rank the elements of V, since clearly the cardinality of ord(V) is double than that of V.