Symposium · Logic & Paradoxes · Pl. 11Bertrand Russell · 1872–1970
Russell's Paradox
A crack in the foundations
Around 1901, working through the logical foundations mathematicians were busy building for their entire discipline, Russell found a single, devastatingly simple question that the whole system could not answer consistently. It concerned nothing more exotic than classes, groups of things, and whether a class could be a member of itself. The question fit in a sentence, and the contradiction followed in two lines. Russell spent the next decade, with Whitehead, building a system in which the question could not be asked.
The world it came from
Russell found the contradiction in the spring of 1901, working on The Principles of Mathematics and trying to derive all of mathematics from logic. Cantor's set theory was two decades old and already had a crack, the Burali-Forti paradox of 1897 about the largest ordinal, but Russell's was simpler and struck at the basic assumption that any condition picks out a class. On 16 June 1902 he wrote to Gottlob Frege, whose Grundgesetze rested on exactly that assumption and whose second volume was at the printer. Frege replied that arithmetic itself seemed to totter, and added an appendix admitting the flaw. Zermelo had found the same contradiction independently in Göttingen. Russell's repair, the theory of types, appeared in 1908 and was built into Principia Mathematica from 1910.
The idea
Form the assemblage of all classes which are not members of themselves. Is it a member of itself or not? If it is, it is one of those classes that are not members of themselves, so it is not a member of itself. If it is not, it is one of those classes that are not members of themselves, so it is a member of itself. Each answer proves its own opposite. The question is legitimate on the face of it. A class either contains itself or it does not. This one cannot do either.
How the reasoning runs
Premise one, naive comprehension: for any condition you can state, there is a class of exactly the things that meet it. Premise two, the condition: is not a member of itself. By premise one, form R, the class of all classes that are not members of themselves. Ask whether R is a member of R. Suppose it is. Then R meets the condition for membership in R, so R is not a member of itself. Suppose it is not. Then R fails the condition, which means R is a member of itself. So R is a member of itself if and only if it is not. The contradiction bites at premise one. Some conditions pick out no class at all.
Seen in life
You are asked to build the company's master index: a page listing every internal page that does not list itself. Most pages qualify; the holiday policy does not link to itself. You create the index and start adding entries. Then someone asks whether the index should appear in the index. If it does not list itself, it qualifies, so it should. Add the line, and now it lists itself, so it no longer qualifies and the line must go. The page can never be finished, and the problem is not the software. The specification named a collection that cannot exist. The barber who shaves everyone who does not shave himself has the same shape, and the same answer: there is no such barber, and there is no such page.
Why it matters
This was not a minor technical glitch. It struck at Gottlob Frege's foundational project for mathematics just as it was nearing completion, and forced logicians to rebuild set theory from the ground up with careful restrictions, the theory of types among them, on what counts as a legitimate class in the first place. Frege wrote that a scientist can hardly meet anything less welcome than to have the foundation give way just as the work is finished. The remark has outlived the system it mourned.
“Form now the assemblage of all classes which are not members of themselves. This is a class: is it a member of itself or not?”
From the source
Russell states the contradiction directly: “Form now the assemblage of all classes which are not members of themselves. This is a class: is it a member of itself or not?... Thus of the two hypotheses—that it is, and that it is not, a member of itself—each implies its contradictory. This is a contradiction.” The passage comes from Chapter XIII, on the axiom of infinity and logical types, where Russell explains why classes cannot be treated as ordinary objects. He argues that the paradox, like the Liar, comes from a vicious circle: a collection defined in terms of a totality that includes the thing being defined. His remedy is to arrange classes in levels, so that the question of self-membership cannot be asked.
The pushback
Doesn't restricting which classes are allowed to exist, simply to dodge this one contradiction, look like patching the theory rather than genuinely explaining what went wrong with the original, seemingly innocent idea of a class? Russell's own theory of types answers by treating levels of classes as a real logical structure, not an arbitrary fix, though the debate over how well it succeeds continues. A second objection: the theory of types makes mathematics unbearably awkward, with a separate number two at every level. Principia needed an axiom of reducibility to get around this, and nobody thought that axiom was a truth of logic.
The reply
Mathematics chose a different repair. Ernst Zermelo's axioms of 1908, extended by Fraenkel and Skolem in the 1920s, replaced naive comprehension with separation: you may carve a subclass out of a set you already have, but you may not conjure a set from a condition alone. In ZF, Russell's R is not a set, and the contradiction becomes a short proof that no set contains everything. Von Neumann kept a version of R as a proper class, a collection too large to be a member of anything. What no repair supplies is an explanation of why the naive idea was wrong. The iterative conception, on which sets are built in stages from below, is the standard picture, but the debate over it continues.
Where it went next
Frege's system fell and Principia Mathematica rose in its place, three volumes that Gödel then used, in 1931, as the target system for his incompleteness theorems. Zermelo-Fraenkel set theory became the working foundation of mathematics. Russell's types became Alonzo Church's typed lambda calculus in 1940 and, from there, the type systems of programming languages: a compiler that refuses to let a list contain itself as an element is enforcing Russell's levels. Cantor's diagonal argument, the paradox's ancestor, reappears in Turing's proof that no program can decide whether every program halts. The barber version, which Russell attributed to someone else, is how most people first meet the idea.
Where this leads
Consider 'the present King of France is bald' shows the very same careful logician turning his attention to a smaller, related puzzle: how ordinary language quietly assumes things exist that logic has no business simply taking for granted at all. The Liar is the older cousin, and Russell thought them one family. In each, something refers to itself and the reference flips its own value. His theory of types was meant to forbid both at once, and the comparison shows what he was really after: not a fix for classes, but a ban on a shape of thought.