# Zermelo-Frankl axioms

## Summary

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.

## Context

This concept has the prerequisites:

- set operations (The Z-F axioms are a formalism for set theory.)
- Russell's Paradox (Russell's Paradox motivates the need for axiomatic set theory.)
- first-order logic (The Z-F axioms are typically defined in first-order logic.)
- natural numbers as sets (The interpretation of natural numbers as sets is needed to motivate the Axiom of Infinity.)

## Goals

- 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)

## -Free-

→ 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

Other notes:

- Skim Section 4, "Ordinals and their basic properties," for the Axiom of Foundation.

## -Paid-

→ A Course in Mathematical Logic

A graduate textbook in mathematical logic.

Location:
Section 10.1, "Basic developments," pages 459-468

Other notes:

- Skim Section 10.3 for the Axiom of Regularity.

→ Elements of Set Theory

An introductory textbook on axiomatic set theory.

- Chapter 2, "Axioms and operations," pages 17-33
- Chapter 4, "Natural numbers," subsection "Inductive sets," pages 67-70

Other notes:

- These sections don't cover the Axioms of Replacement and Regularity; you need to look those up in the index.

## See also

-No Additional Notes-