{'fraction_non_alphanumeric': 0.0188929400066291, 'fraction_numerical': 0.008286377195889956, 'mean_word_length': 4.988095238095238, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
The Dividing Line Methodology: Model Theory Motivating Set Theory

The 1960’s produced technical revolutions in both set theory and model theory. Researchers such as Martin, Solovay, and Moschovakis kept the central philosophical importance of the set theoretic work for the foundations of mathematics in full view. In contrast the model theoretic shift is often seen as ‘ technical’ or at least ‘merely mathematical’. Although the shift is productive in multiple senses, is a rich mathematical subject that provides a metatheory in which to investigate many problems of traditional mathematics: the profound change in viewpoint of the nature of model theory is overlooked. We will discuss the effect of Shelah’s dividing line methodology in shaping the last half century of model theory. This description will provide some background, definitions, and context for [She19]. In this introduction we briefly describe the paradigm shift in first order1 model theory that is laid out in more detail in [Bal18]. We outline some of its philosophical consequences, in particular concerning the role of model theory in mathematics. We expound in Section 1 the classification of theories which is the heart of the shift and the effect of this division of all theories into a finite number of classes on the development of first order model theory, its role in other areas of mathematics, and on its connections with set theory in the last third of the 20th century. We emphasize that for most practitioners of late 20th century model theory and especially for applications in traditional mathematics the effect of this shift was to lessen the links with set theory that had seemed evident in the 1960’s. In Section 2 we explore how Shelah’s underlying methodological precept of dividing lines led to the refinement of this classification to admit infinitely many classes, deeper connections with set theory, and to the current emphasis on ‘neo-stability theory’ in both pure and applied model theory. These developments undermine the impression from Section 1 that classification theory completely freed first order model theory from considering extensions of ZFC. In Section 3, we build on Maddy’s [Mad19] discussion of the role of foundations to explore the ways in which model theory can provide essential guidance in both traditional mathematics and for axiomatic set theory. What I call the paradigm shift in model theory [Bal18] took place in two phases. The first phase is a shift from the Russell-Hilbert-Gödel conception of higher-order logic as a general framework for all of mathematics to a Robinson-Tarski focus on first order theories to study distinct areas of mathematics. One key to this shift is the switch from a logic that allows quantification over predicate variables of all arities and all orders to the modern conception of fixing a vocabulary with a fixed set τ of relation symbols relevant to the area being studied and quantifying only over individuals. The focus becomes, not the study of logic(s),