---
tag: 7FW29X1ADK
kind: statement
role: proposition
name: Gödel's axiom system (NBG)
sources:
  - wikipedia: en:528491@r1330985339
    locator: Von Neumann–Bernays–Gödel set theory, section History > Gödel's axiom system (NBG)
    note: adopted
    mention: wikipedia://en/528491#M8BC
    sections: [Statement]
inventory: [SR48KHNH57]
---

## Statement

In 1931, Bernays sent a letter containing his set theory to Kurt Gödel. Gödel simplified Bernays' theory by making every set a class, which allowed him to use just one sort and one membership primitive. He also weakened some of Bernays' axioms and replaced von Neumann's choice axiom with the equivalent axiom of global choice. Gödel used his axioms in his 1940 monograph on the relative consistency of global choice and the generalized continuum hypothesis.

Several reasons have been given for Gödel choosing NBG for his monograph:

- Gödel gave a mathematical reason—NBG's global choice produces a stronger consistency theorem: "This stronger form of the axiom \[of choice\], if consistent with the other axioms, implies, of course, that a weaker form is also consistent."
- Robert Solovay conjectured: "My guess is that he \[Gödel\] wished to avoid a discussion of the technicalities involved in developing the rudiments of model theory within axiomatic set theory."
- Kenneth Kunen gave a reason for Gödel avoiding this discussion: "There is also a much more combinatorial approach to **L** \[the constructible universe\], developed by ... \[Gödel in his 1940 monograph\] in an attempt to explain his work to non-logicians. ... This approach has the merit of removing all vestiges of logic from the treatment of **L**."
- Charles Parsons provided a philosophical reason for Gödel's choice: "This view \[that 'property of set' is a primitive of set theory\] may be reflected in Gödel's choice of a theory with class variables as the framework for ... \[his monograph\]."

Gödel's achievement together with the details of his presentation led to the prominence that NBG would enjoy for the next two decades. In 1963, Paul Cohen proved his independence proofs for ZF with the help of some tools that Gödel had developed for his relative consistency proofs for NBG. Later, ZFC became more popular than NBG. This was caused by several factors, including the extra work required to handle forcing in NBG, Cohen's 1966 presentation of forcing, which used ZF, and the proof that NBG is a conservative extension of ZFC.