We introduce indecomposable (hooked) Skolem sequences and we show their existence for all admissible orders. We then introduce a new construction for two-fold triple systems from two-fold Skolem sequences, and use this to motivate the concept of a two-fold Rosa sequence, whose existence we show for all admissible orders.

Expansion trees are defined as generalizations of Herbrand instances for formulas in a nonextensional form of higher-order logic based on Church’s simple theory of types. Such expansion trees can be defined with or without the use of skolem functions. These trees store substitution terms and either critical variables or skolem terms used to instantiate quantifiers in the original formula and th...