- fundamentals of set theory
- material set theory
- presentations of set theory
- structuralism in set theory
- class-set theory
- constructive set theory
- algebraic set theory

Given a set $V$ with an extensional relation $\prec$, a **proto-powerset structure** is a function $\mathcal{P}:V \to V$ such that

- for all elements $b \in V$, and $c \in V$, if for all elmemets $a \in V$, $a \prec b$ implies $a \prec c$, then $b \prec \mathcal{P}(c)$.

A **powerset structure** is a function $\mathcal{P}:V \to V$ such that

- for all elements $b \in V$, and $c \in V$, $b \prec \mathcal{P}(c)$ if and only if for all elements $a \in V$, $a \prec b$ implies $a \prec c$.

Uniqueness of $\mathcal{P}(c)$ follows from $\prec$ being an extensional relation.

In any material set theory, instead of postulating the mere existence of a set $\mathcal{P}$ in which for all sets $b$, if for all sets $a$, $a \in b$ implies $a \in c$, then $b \in \mathcal{P}$, one could add a primitive unary operation $\mathcal{P}(c)$ which takes material sets $c$ and returns a material set $\mathcal{P}(c)$ such that for all $b$, if for all $a$, $a \in b$ implies $a \in c$, then $b \in \mathcal{P}(c)$.

Last revised on December 12, 2022 at 18:06:31. See the history of this page for a list of all contributions to it.