We reported in the introduction (page 15) several questions raised by Mic Detlefsen. In less precise fashion, they were: Is it better for a theory to be categorical or not? How do we justify whether categoricity is a virtue for a theory? Is completeness a good approximation to categoricity? This part has several themes that together respond to these questions.

Chapters 1 and 2 lay out the basic mathematical and philosophical (respectively) terminology of this book. Chapters 1.1 and 1.2 primarily address theme (1). Chapter 1.3 clarifies the role of various logics while properties of theories and axioms are examined in Chapter 1.4. Chapter 2 addresses philosophical issues about our notion of formalization. First we stress that it is a process and then we distinguish two possible goals: foundational and instrumental. Chapter 2.3 expounds the criterion of theme (3). Chapter 2.4 outlines how these virtuous properties can serve as organizing principles for mathematics and introduces the stability hierarchy, a set of virtuous properties that provide a specific method for such an organization that has powerful consequences for finding invariants for models.

These distinctions underlie the argument in Chapters 3.1 and 3.2, which deal with theme (2). Chapters 3.3 and 3.4 develop the notion of categoricity in power, whose powerful consequences in finding invariants for models signal the importance of studying classes of theories. Thus we initiate the study of the paradigm shift which occupies Part II.