# Russell's Paradox

## Summary

Consider the set S of all sets which are not members of themselves: is S a member of itself? Either answer leads to a contradiction. Russell's Paradox, as this is called, showed that Cantor's original formulation of set theory was inconsistent and led to more precise axiomatizations of set theory, such as the Zermelo-Frankl axioms.

## Context

This concept has the prerequisites:

- set operations (Russell's paradox is an inconsistency in naive set theory.)
- first-order logic (The Axiom of Comprehension is expressed in first-order logic.)

## Goals

- Know what Russell's Paradox refers to and why it is a contradiction in naive set theory

- Know what is meant by the (unrestricted) Axiom of Comprehension and why it is responsible for the paradox

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

## -Free-

→ Wikipedia

→ An Introduction to Set Theory (2008)

Lecture notes on axiomatic set theory.

## -Paid-

→ The Language of First-Order Logic

An undergraduate logic textbook aimed at philosophers, with an educational software package.

- Section 8.1, "Cantor's set theory," pages 208-211
- Section 8.7, "Russell's Paradox," pages 220-222

## See also

-No Additional Notes-