May 5, 2025

Umma Habiba Begum 

11th Grade

The Young Women's Leadership School of Queens

Have you encountered such logical riddles where the answers are both right and wrong simultaneously? A classic example is the Liar's Paradox—a statement that claims to be false. Now, thinking logically, if the statement is true, it is false, and if it is false, it is true. This is known as a paradox puzzle.

In the 20th century, a similar paradox shook the mathematical world, which involved set theory. It revealed a contradiction in how sets were understood, particularly for "naive set theory." Naive set theory is an informal way for us to understand, using ordinary language, what sets are and how they are used. This relates to another method of using a made-up language, similar to what is used in computing and coding, called an axiomatic system. Bertrand Russell was a British philosopher, logician, and mathematician—one of the most influential thinkers of the 20th century—and he discovered a significant flaw, particularly when analyzing Cantor’s power class theorem of set theory, while reading Gottlob Frege’s work about set theory and how it was a foundation of all mathematics. For instance, Set R is the set of all sets that do not contain themselves. If R contains itself, it breaks its own rule, so it should not contain itself. However, if R does not contain itself, it fits the rule, so it should contain itself. So it is a contradiction all around.

In 1902, he wrote a letter to Gottlob Frege to relay his findings, to which Frege responded, acknowledging the problem. Despite recognizing the issue, he published his book about set theory, including some modifications, a statement, and an attempt to revise his system to avoid the contradiction. Nonetheless, his revised system was later also shown to be flawed, leading him to abandon it. 

“A scientist can hardly meet with anything more undesirable than to have the foundations give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press.” -- Gottlob Frege

Consistency is essential in mathematics; therefore, definitions and rules must be strict to have a proper answer without contradiction. This led many different mathematicians to rethink and put restrictions on set theory in order to prevent a possible contradiction. For example, Russell himself created a type theory that prevented sets from containing themselves by organizing them into levels or "types," which blocks self-reference; more famously, the Zermelo-Fraenkel Set Theory (ZF/ZFC), which uses strict axioms/rules that are "obviously true" without requiring any proof, to avoid forming problematic sets like the one in the paradox. Overall, Russell's Antinomy or paradox established that previously known knowledge should be cautious when there are inconsistencies and refined.