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.
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.)
- 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)
→ An Introduction to Set Theory (2008)
Lecture notes on axiomatic set theory.
→ 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
-No Additional Notes-
- create concept: shift + click on graph
- change concept title: shift + click on existing concept
- link together concepts: shift + click drag from one concept to another
- remove concept from graph: click on concept then press delete/backspace
- add associated content to concept: click the small circle that appears on the node when hovering over it
- other actions: use the icons in the upper right corner to optimize the graph placement, preview the graph, or download a json representation