Maths WorkshopMaths problems with full solutions
Exercise 1761Logic, sets and proofs › Proof techniquesL1Added 3 Nov 2026

Russell's paradox (statement)

#Russell paradox#sets#self-reference

Statement

State Russell's paradox and explain its consequence for naive set theory.

Hint

Consider the "set of all sets not containing themselves".

Show solutionHide solution

Solution

Paradox. Let R={S:S∉S}R=\{S:S\notin S\}. Then R∈R  ⟺  R∉RR\in R\iff R\notin R, contradiction.

Consequence. Naive comprehension (every property defines a set) is inconsistent. Axiomatic set theory (ZFC) restricts comprehension to avoid self-reference.

Conclusion. Russell's paradox motivated modern axiomatic set theory.

More problems in Proof techniques

Ex.248Proof by counterexample
#counterexample #disproof #universal statement
Proof techniques
High school
High school
Ex.249De Morgan's laws
#De Morgan #logic #negation
Proof techniques
High school
High school
Ex.250Direct proof
#direct proof #implication #even odd
Proof techniques
High school
High school
Ex.495Proof by contradiction
#contradiction #irrationality #sqrt2
Proof techniques
High school
High school