On the distinction between sets and classes

a categorical perspective

Samuele Maschio

pp. 185-199

in: Gabriele Lolli, Marco Panza, Giorgio Venturi (eds), From logic to practice, Berlin, Springer, 2015

Abstract

According to Kreisel, category theory provides a powerful tool to organize mathematics. An example of this descriptive power is the categorical analysis of the practice of presenting 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), the mathematical level can be represented by internal categories. Through this two-level interpretation, we can clarify the relation between classes and sets in ZF; in particular, we can describe two equivalent categorical notions of definable sets. Some common sayings about set theory are interpreted on the basis of this representation, emphasizing the distinction between naïve and rigorous sentences about sets and classes.