A new class of order types

Abstract Let φ 4 be the class of all order-types ϕ with the properties that every uncountable subtype of ϕ contains an uncountable well-ordering, but ϕ is not the union of countably many well-orderings. It is proved that φ 4 ≠ 0, and a way is found of associating stationary sets with most of the types in φ 4 which is useful for applications. A number of results concerning the structure and embeddability properties of φ 4 are obtained, including some consistency and independence results. One consequence is the independence of Jensen's combinatorial principle □ ω 1 .