Zermelo-Frankl axioms


The Zermelo-Frankl axioms are the standard formalization of set theory in mathematics. They include the Axiom of Extensionality (two sets are equal if they have the same elements), various comprehension axioms (which enable one to construct sets), and the Axiom of Regularity (which rules out certain pathological sets). Most of mathematics can be derived from these axioms.


This concept has the prerequisites:


  • Know the Axiom of Extensionality
  • Know the various comprehension axioms (e.g. pairing, union, power set, subset, replacement, infinity; specific presentations will differ)
  • Know the Axiom of Foundation (or Axiom of Regularity) and what pathological sets it is meant to rule out
  • Be able to derive simple consequences of the axioms
  • What is the difference between a set and a class?

Core resources (read/watch one of the following)


Notes on Set Theory (2013)
Lecture notes for a course on axiomatic set theory.
Location: Section 1, "First axioms of set theory," pages 1-7
Author: Henry Cohn
Other notes:
  • Skim Section 4, "Ordinals and their basic properties," for the Axiom of Foundation.


See also

-No Additional Notes-