Can a Set Contain Itself? A Question That Broke Mathematics

Suppose collections can be defined by any property, which is the natural assumption and the one early set theory made. Consider the collection of all collections that are not members of themselves. Ask whether that collection is a member of itself. If it is, then by its own defining property it must not be, since it contains only collections that are not members of themselves. If it is not, then it satisfies the defining property and therefore must be a member. Either answer contradicts itself immediately, which means no such collection can exist, which means the assumption that any property defines a collection is false. That assumption was the foundation of the subject.

The contradiction struck at a specific project at a specific moment:

• •Set theory was being used to provide secure foundations for the whole of mathematics
• •Gottlob Frege had spent years building arithmetic on exactly the assumption the contradiction destroys
• •Bertrand Russell wrote to Frege in 1902 while the second volume was at the printer
• •Frege added an appendix acknowledging that the foundation of his work was undermined
• •Similar contradictions were found in other formulations shortly afterwards
• •The episode is among the most documented moments in the history of mathematics

The response was to restrict which collections are allowed to exist rather than to abandon the subject. Axiomatic set theories were developed specifying exactly what may be formed, with the crucial change being that a property does not by itself define a set, and a set can only be formed by selecting from a set that already exists, which blocks the construction entirely. The system that became standard, developed by Zermelo and refined by Fraenkel and others, has served since without producing a contradiction, which is reassuring and is not a proof that none exists. Alternative responses included Russell's own theory of types, which forbids a collection from being applied to itself by arranging everything in levels, and it influenced the design of programming languages substantially.

Several related contradictions arise from self-reference and comparing them clarifies what the problem is. The liar paradox, considering a sentence asserting its own falsity, is ancient and has the same structure. The barber who shaves everyone who does not shave themselves cannot consistently either shave himself or not, which is the set paradox restated in ordinary language and is easier to dismiss because no such barber need exist. A paradox about the collection of all collections asks whether it contains itself. One about naming considers the smallest number not describable in fewer than a stated quantity of words, which the description itself then describes. What these share is a construction referring to a totality it is itself part of, which is what the modern restrictions block.

The episode opened a broader question about whether mathematics could be secured at all, and the answer turned out to be discouraging. Hilbert proposed a programme to prove that a formal system of arithmetic is consistent, meaning free of contradiction, using only methods the system itself contains. Kurt Gödel showed in 1931 that this cannot be done, and that any consistent system strong enough to express arithmetic contains true statements it cannot prove. That result is not a paradox and is frequently misused as though it licensed vagueness about anything. What it establishes is precise and narrow, namely that the particular kind of complete self-contained certainty Hilbert wanted is unavailable, which is a genuine limit rather than a reason to doubt arithmetic.

The collection of all collections not containing themselves cannot consistently either contain itself or not, which shows that a property does not by itself define a collection. Russell put it to Frege in 1902 while the second volume of his foundational work was at the printer. The resolution restricts which sets may be formed, and Gödel later showed that a system cannot prove its own consistency.