The first of NASA's high-level strategic goals is to extend and sustain human activities across the solar system. As the United States moves into the post-Shuttle era, meeting this goal is more challenging than ever. There are several desired outcomes for this goal, including development of an integrated architecture and capabilities for safe crewed and cargo missions beyond low Earth orbit. NASA's Flexible Path for the future human exploration of space provides the guidelines to achieve this outcome. Designing space system architectures to satisfy the Flexible Path starts early in design, when a downselection process works to reduce the broad spectrum of feasible system architectures into a more refined set that contains a handful of alternatives that are to be considered and studied further in the detailed design phases. This downselection process is supported by what is referred to as architecture space exploration (ASE). ASE is a systems engineering process which generates the design knowledge necessary to enable informed decision-making. The broad spectrum of potential system architectures can be impractical to evaluate. As the system architecture becomes more complex in its structure and decomposition, its space encounters a factorial growth in the number of alternatives to be considered. This effect is known in the literature as combinatorial explosion. For the Flexible Path, the development of new space system architectures can occur over the period of a decade or more. During this time, a variety of changes can occur which lead to new requirements that necessitate the development of new technologies, or changes in budget and schedule. Developing comprehensive and quantitative design knowledge early during design helps to address these challenges. Current methods focus on a small number of system architecture alternatives. From these alternatives, a series of 'one off' -type of trade studies are performed to refine and generate more design knowledge. These small-scale studies are unable to adequately capture the broad spectrum of possible architectures and typically use qualitative knowledge. The focus of this research is to develop a systems engineering method for system-level ASE during pre-phase A design that is rapid, exhaustive, flexible, traceable, and quantitative. Review of literature found a gap in currents methods that were able to achieve this research objective. This led to the development of the Set Theory-Influenced Architecture Space Exploration (STASE) methodology. The downselection process is modeled as a decision-making process with STASE serving as a supporting systems engineering method. STASE is comprised of two main phases: system decomposition and system synthesis. During system decomposition, the problem is broken down into three system spaces. The architecture space consists of the categorical parameters and decisions that uniquely define an architecture, such as the physical and functional aspects. The design space contains the design parameters that uniquely define individual point designs for a given architecture. The objective space holds the objectives that are used in comparing alternatives. The application of set theory across the system spaces enables an alternative form of representing system alternatives. This novel application of set theory allows the STASE method to mitigate the problem of combinatorial explosion. The fundamental definitions and theorems of set theory are used to form the mathematical basis for the STASE method. A series of hypotheses were formed to develop STASE in a scientific way. These hypotheses are confirmed by experiments using a proof of concept over a subset of the Flexible Path. The STASE method results are compared against baseline results found using the traditional process of representing individual architectures as the system alternatives. The comparisons highlight many advantages of the STASE method. The greatest advantage is that STASE comprehensively explores the architecture space more rapidly than the baseline. This is because the set theory-influenced representation of alternatives has a summation growth with system complexity in the architecture space. The resultant option subsets provide additional design knowledge that enables new ways of visualizing results and comparing alternatives during early design. The option subsets can also account for changes in some requirements and constraints so that new analysis of system alternatives is not required. An example decision-making process was performed for the proof of concept. This notional example starts from the entire architecture space with the goal of minimizing the total cost and the number of launches. Several decisions are made for different architecture parameters using the developed data visualization and manipulation techniques until a complete architecture was determined. The example serves as a use-case example that walks through the implementation of the STASE method, the techniques for analyzing the results, and the steps towards making meaningful architecture decisions.