# 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:

- set operations
- equivalence relations (One needs to define equality.)
- order relations (One needs to define the comparison operators.)
- Peano axioms (One must show that the construction satisfies the Peano axioms.)

## 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.

## Core resources (read/watch one of the following)

## -Paid-

→ Elements of Set Theory

An introductory textbook on axiomatic set theory.

Location:
Chapter 4, "Natural numbers," pages 66-88

## Supplemental resources (the following are optional, but you may find them useful)

## -Free-

→ Notes on Set Theory (2013)

Lecture notes for a course on axiomatic set theory.

→ An Introduction to Set Theory (2008)

Lecture notes on axiomatic set theory.

Additional dependencies:

- Zermelo-Frankl axioms

## See also

-No Additional Notes-