What is the real category of sets?

TL;DR

This paper uses category theory to analyze the concept of sets and classes in ZF set theory, clarifying their relation through a two-level categorical interpretation that distinguishes between naive and rigorous perspectives.

Contribution

It introduces a categorical framework to differentiate between classes and sets in ZF, providing two equivalent notions of definable sets and clarifying foundational concepts.

Findings

Categorical analysis clarifies the relation between classes and sets in ZF.

Two equivalent categorical notions of definable sets are presented.

Interpretation of set theory sayings highlights distinctions between naive and rigorous views.

Abstract

Category theory provides a powerful tool to organize mathematics. A sample of this descriptive power is given by the categorical analysis of the practice of "classes as shorthands" in ZF set theory. In this case category theory provides a natural way to describe the relation between mathematics and metamathematics: if metamathematics can be described by using categories (in particular syntactic categories), then the mathematical level is represented by internal categories. Through this two-level interpretation we can clarify the relation between classes and sets in ZF and, in particular, we can present two equivalent categorical notions of definable set. Some common sayings about set theory will be interpreted in the light of this representation, emphasizing the distinction between naive and rigorous sentences about sets and classes.