Barber paradox - Wikipedia

Abstraction: Self-referential logical paradox derived from Russell's paradox

Key points:

• The barber is defined as "one who shaves all those, and only those, who do not shave themselves" — leading to an irresolvable contradiction about whether the barber shaves himself
• Russell himself did not consider it a valid instance of his own paradox; he argued the question is simply a vacuous false proposition (no such barber can exist)
• Formally, the existential clause is unsatisfiable because the universal quantifier applied to the barber himself produces a direct contradiction P↔¬P
• Russell's original paradox targeted naive set theory as used by Cantor and Frege; the barber variant illustrates the concept accessibly
• Related to Gödel's incompleteness theorems, the halting problem, and the principle of explosion

Connections: Bertrand Russell · Georg Cantor · Gottlob Frege · Logic Paradoxes · Set Theory · Self Reference

Source: https://en.wikipedia.org/wiki/Barber_paradox