2025-03-07-RusselsParadox

Can we just define a universal set with everything in it?

The Problem with Universal Sets

Question: Why can’t we just define a universal set as “the set of everything”?

Answer: Russell’s Paradox shows this leads to contradiction.

The Paradox Setup

Define $S$ as the set that contains all sets that are not elements of themselves:

$S=A\in \Omega :A\notin A$

Where $\Omega$ represents some universal collection.

Most ordinary sets are elements of $S$. For example:

• The set $1,2,3$ is in $S$ because $1,2,3\notin 1,2,3$

The Contradiction

Key Question: Is $S$ an element of itself?

Case 1: If $S\in S$

• By definition of $S$, we must have $S\notin S$
• Contradiction!

Case 2: If $S\notin S$

• By definition of $S$, we must have $S\in S$
• Contradiction!

Attempted Resolution

Try to resolve this by assuming a universal set $U$ exists and redefine:

$S=A\subseteq U:A\notin A$

But the same contradiction arises:

• If $S\in S$, then $S\notin S$ (contradiction)
• If $S\notin S$, then $S\in S$ (contradiction)

Conclusion

Since both cases lead to contradiction, we must conclude that $S⊈U$.

This means: No universal set $U$ can exist, because if it did, then $S$ (which should be a subset of $U$) would lead to Russell’s Paradox.

The paradox reveals fundamental problems with naive set theory and shows why careful axiomatization of set theory is necessary.