natural numbers as sets


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.


This concept has the prerequisites:


  • 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)


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


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-