---
tag: QH884Y0KHC
kind: statement
role: proposition
name: Bernays' axiom system
sources:
  - wikipedia: en:528491@r1330985339
    locator: Von Neumann–Bernays–Gödel set theory, section History > Bernays' axiom system
    note: adopted
    mention: wikipedia://en/528491#5IPW
    sections: [Statement]
inventory: [40BGC2NT83]
---

## Statement

In 1929, Paul Bernays started modifying von Neumann's new axiom system by taking classes and sets as primitives. He published his work in a series of articles appearing from 1937 to 1954. Bernays stated that: The purpose of modifying the von Neumann system is to remain nearer to the structure of the original Zermelo system and to utilize at the same time some of the set-theoretic concepts of the Schröder logic and of *Principia Mathematica* which have become familiar to logicians. As will be seen, a considerable simplification results from this arrangement. Bernays handled sets and classes in a two-sorted logic and introduced two membership primitives: one for membership in sets and one for membership in classes. With these primitives, he rewrote and simplified von Neumann's 1929 axioms. Bernays also included the axiom of regularity in his axiom system.