Trade-offs between design goals have traditionally been studied using optimization approaches, which assume a fixed formulation and framing of the design problem. We propose a novel set-theory framework of design spaces to model the role of problem reformulation and reframing in resolving design trade-offs. The framework predicts mechanisms by which the designer can alter the boundaries and structure of that space to alter or avoid Pareto frontiers in the original space. Empirical evidence from interviews with designers identifies eight distinct trade-off response mechanisms aligned with the framework's predictions. The framework and interview results provide a foundation for developing methodologies that encourage design space restructuring to avoid unnecessary design compromises and sacrifices. • Trade-offs between design goals are traditionally studied as optimization problems. • We use set theory to model role of designer's problem reformulation and reframing. • The framework predicts how the designer can alter the structure of the design space. • We collect interview evidence from designers that have faced trade-off situations. • We identify eight trade-off responses that align with the framework's predictions.