Universe Set

In set theory, the universe is a set that contains all other sets. It's also known as the universal set. $$ U = \{ A, B, C, ... \} $$

The Barber Paradox

The concept of a universe set, however, leads to contradictions.

To explain, let's turn to the barber paradox proposed by Bertrand Russell in the 19th century.

In a town, the barber shaves everyone who does not shave themselves.

Who shaves the barber?

• If the barber shaves himself, then he should not shave himself.
• If the barber does not shave himself, then he should shave himself.

Similarly, one might ask if the universal set U belongs to itself, the universal set U.

In theory, the universal set contains all other sets.

• If U ∈ U, then the set U should be different from U.
• If U ∉ U, then it should belong to U.

Note. In the case of the universal set, I use the membership relation ∈ (not the subset relation ⊆) because the elements of the universal set are other sets. It is, therefore, correct to write A∈U.

Here's another example to further explain.

A set S contains all sets that do not contain themselves.

• If S ∈ S, then it should not contain itself.
• If S ∉ S, then it should contain itself.

It's another contradiction.

How can we resolve these contradictions?

To solve these issues, one can define the universal set U as the set that contains all the sets one is working with.

This narrows the scope to only the sets in reference.

All others are not considered.