Download PDFOpen PDF in browserGenerating Custom Set Theories with NonSet Structured ObjectsEasyChair Preprint 566317 pages•Date: June 1, 2021AbstractSet theory has long been viewed as a foundation of mathematics, is pervasive in mathematical culture, and is explicitly used by much written mathematics. Because arrangements of sets can represent a vast multitude of mathematical objects, in most set theories every object is a set. This causes confusion and adds difficulties to formalising mathematics in set theory. We wish to have set theory's features while also having many mathematical objects not be sets. A generalized set theory (GST) is a theory that has pure sets and may also have nonsets that can have internal structure and impure sets that mix sets and nonsets. This paper provides a GSTbuilding framework. We show example GSTs that have sets and also (1) nonset ordered pairs, (2) nonset natural numbers, (3) a nonset exception object that can not be inside another object, and (4) modular combinations of these features. We show how to axiomatize GSTs and how to build models for GSTs in other GSTs. Keyphrases: abstraction, foundations of mathematics, inductive datatypes, interactive theorem proving, set theory
