Set Theory Paradoxes

Set theory paradoxes are contradictory situations that arose in naive set theory. These paradoxes showed that not every arbitrarily formulated set can be considered existent, and a stricter axiomatic system is needed (for example, Zermelo–Fraenkel set theory).

Russell's Paradox

Russell's paradox starts from the following question: consider those sets that do not contain themselves as elements. Let R = { A | A ∉ A }. Now, ask the question: is R ∈ R?

• If R ∈ R, then by definition, R should not contain itself → contradiction.
• If R ∉ R, then by definition, R should contain itself → also contradiction.

This contradiction points out that the naive set theory is unregulated, and not every 'rule'-defined set can be considered valid.

Barber Paradox

The everyday equivalent of Russell's paradox is the Barber paradox: in a village, the barber shaves exactly those who do not shave themselves. The question: who shaves the barber?

• If the barber shaves himself, then he shouldn't shave himself.
• If he doesn't shave himself, then he should shave himself.

This is also a contradiction that well illustrates the essence of Russell's paradox in everyday life.

Lesson

The paradoxes pointed out that set theory must be regulated with axioms. As a result, Zermelo–Fraenkel set theory (ZF) emerged, which provides a safer framework for modern mathematics.

Solved Example

Suppose a set is defined as: S = { x | x is a set, and x ∉ x }. Decide if S ∈ S!

• If S ∈ S, then by definition, S cannot be its own element.
• If S ∉ S, then by definition, S should contain itself.

Both cases lead to contradiction. This is exactly the logic of Russell's paradox.

Practice Exercise