📂Set Theory

Russell's Paradox

Paradox ¹

If there exists a set of all sets $\mathscr{U}$, then there exists some set $R$ that is both a member of $\mathscr{U}$ and not a member of itself.

Description

In the 6th century BC, the philosopher from Crete, Epimenides, declared:

“All Cretans are liars!”

If Epimenides’ statement is true, then, being a Cretan himself, his statement would be false. However, if his statement is false, then Epimenides is a liar, which means his statement doesn’t contradict the fact and is therefore true. This is why the word ‘all’ is so perilous in logic.

From 1874 to 1884, the studies of Cantor were the foundation of what would later be known as set theory. At that time, the academic backlash was so severe that it nearly drove him to madness, but by the time Bertrand Russell published this paradox in 1902, Cantor’s set theory had already become a fundamental part of mathematics. Though Cantor’s life was tragic, it took only about 20-30 years for his theory to engulf the academic world. The great mathematician Hilbert said, ‘No one shall expel us from the paradise that Cantor has created.’ It has become difficult to imagine rigorous mathematics without the concept of sets.

The announcement of Russell’s Paradox shook the very foundation of mathematics at the time. What if the contradiction found by Russell exists in the set one is working with? If the conditions causing such a paradox are not identified, all research will always carry an element of instability. The rigor, which mathematics takes pride in and loves, would be shattered from the first line of proof, and any result could potentially harbor the danger of Russell’s Paradox.

Let’s examine how $R:= \left\{ S \in \mathscr{U} : S \notin S \right\}$ creates a problem:

• If $R \in R$, then it’s not $R \notin R$, so $S = R \in \mathscr{U}$ does not meet the condition $(S \notin S)$ and cannot be included in $R$. Thus, $R \notin R$.
• If $R \notin R$, then $R$ meets the condition $(S \notin S)$ and is included in $R$. Therefore, $R \in R$.
• However, according to the law of excluded middle², it cannot be both $R \notin R$ and $R \in R$.

Through this brief argument, we can understand that the premise of the existence of a ‘set of all sets’ is inherently flawed. To ensure this, axioms like the Axiom of Hierarchy or the Axiom of Regularity are deemed necessary, leading scholars to move away from Naive Set Theory towards a more rigorous axiom system.