# natural numbers as sets

## Summary

The natural numbers can be explicitly constructed as sets. Zero is defined to be the empty set, and each n > 0 is defined to be the set of natural numbers less than n. This is an example of how set theory serves as a powerful foundation for much of mathematics.

## Context

This concept has the prerequisites:

## Goals

• Construct the natural numbers in terms of sets.
• Show that the constructed number system satisfies the Peano postulates.
• Define equality and comparison operators, and show that these are an equivalence relation and order relations, respectively.
• Define the addition and multiplication operators.
• Show that these operators are commutative and associative.

## -Free-

Notes on Set Theory (2013)
Lecture notes for a course on axiomatic set theory.
Author: Henry Cohn
An Introduction to Set Theory (2008)
Lecture notes on axiomatic set theory.
Author: William A. R. Weiss
Additional dependencies:
• Zermelo-Frankl axioms

## See also

-No Additional Notes-