If K ⊂ L is an infinite-dimensional algebraic field extension and the (infinite) cardinal number ℵ := [L : K] (the K-vector space dimension of L), then there exists an infinite maximal chain, C, consisting of fields contained between K and L, such that the cardinality of C is at most ℵ. If K ⊂ L is a J-extension, then every maximal chain of intermediate fields has cardinality ℵ0. However, an example is given where K ⊂ L has maximal chains, D and E, of intermediate fields such that the cardinalities of D and E are ℵ and 2ℵ, respectively.