Object generators, categories, and everyday set theory

TL;DR

This paper simplifies the foundational set theory based on object generators, explores properties of the non-set object (like ordinals), and demonstrates applications in category theory, making the framework more accessible for users.

Contribution

It provides a simpler, binary-logic-based description of the object generator framework and explores its implications for set theory and category theory.

Findings

Simplified set theory description using binary logic.

Analysis of the smallest non-set object and its properties.

Application of object generators to natural category theory work.

Abstract

In "Object generators, relaxed sets, and a foundation for mathematics", we introduced ``object generators'', a logical environment much more general than set theory. Inside this we found a `relaxed' version of set theory. That paper is focused on construction of the universal Zermillo-Fraenkel-Choice theory, and the argument that it alone is consistent with mainstream mathematical practice. This paper is oriented toward potential users. The first topic is that if the general context is not needed then there is a simpler description of the set theory. In particular this uses only familiar binary logic, and the resut is almost the same as na\"\i ve set theory. The second topic collects facts about the smallest object that is not a set (the traditional Ordinal numbers, or ``class of all sets''). Quite a bit is known, but it heavily involves non-binary assertion logic. For instance the powerset of this object is the disjoint union of the bounded subobjects, and the cofinal subobjects. However there is no function to {yes, no} that detects this decomposition. The third topic illustrates how the general object-generator context enables natural and full-precision work with categories.