Is There a Set of Everything? The Rival Answers That Both Work
A Letter That Broke Math
In June 1901, a young philosopher named Bertrand Russell (1872–1970) wrote a letter to the famous logician Gottlob Frege (1848–1925). He thanked Frege for his great work, then added a small curiosity: a single question that made Frege’s entire system collapse. It was about sets — simple collections of things, like a box of rocks or a list of your friends.
Before Russell’s letter, most mathematicians believed in a straightforward rule: for any property you could describe, there is a set of all the things that have that property. If you can say “x is red,” there is a set of all red things. If you can say “x is a number bigger than 5,” there is a set of all numbers bigger than 5. This is called naïve comprehension. It felt obvious and harmless.
Russell asked: What about the property “x is a set that does not contain itself”? Most sets don’t contain themselves. The set of all planets is not a planet, so it’s not a member of itself. The set of all spoons is not a spoon. But if the set of all sets that don’t contain themselves exists, does it contain itself? If it does, then it must not (because it only contains sets that don’t contain themselves). If it doesn’t, then it must (because it’s a set that doesn’t contain itself, so it belongs in the collection). Either way you get a contradiction. The whole idea of sets seemed to self-destruct.
Two Different Fixes: Ban Loops or Ban Loose Talk
The paradox showed that you can’t just recklessly collect things into a set. Philosophers and mathematicians had to find a way out. Two rival approaches took the lead.
The first approach, called ZF (after Ernst Zermelo (1871–1953) and Abraham Fraenkel (1891–1965)), decided that the real problem is membership loops. A set can’t be allowed to have an infinite descending chain of memberships, like … a₃ ∈ a₂ ∈ a₁. Sets must be wellfounded — you can always trace membership back to a bottom layer of empty or basic things. In ZF, you build sets step by step from the ground up. You start with the empty set, then make sets of it, then sets of those, and so on. Because no set can contain itself or loop back, you can never build a “set of all sets.” The universal set simply doesn’t exist in ZF. That turns out to be fine: you can still do all of ordinary mathematics with wellfounded sets. The paradoxes disappear, and ZF became the standard foundation for most of the 20th century.
The second approach, NF (New Foundations), took a very different path. Its father, the philosopher W. V. Quine (1908–2000), thought the problem wasn’t membership loops themselves, but how you describe sets. NF’s rule is stratification. Imagine you assign a “level” number to every variable in a formula. The formula is stratified if you can place all the “∈” signs between variables of consecutive levels, like “x₂ ∈ y₃,” and the equals sign only between variables of the same level. If a property can be written in this leveled language, even after erasing the levels, then the set of all things with that property exists. The formula “x is a set and x ∉ x” becomes “x₂ is a set and x₂ ∉ x₂,” which tries to put the same variable on both sides of “∈” with the same level — that’s not allowed. So the Russell set never gets formed. NF blocks the paradox not by banning loops, but by banning the badly structured description.
A Universe of Sets, but at a Price
Because NF allows well-stratified formulas, it lets in the universal set — the set that contains everything, including itself. There’s also a single set of all natural numbers, all real numbers, all groups, all rings, and so on. These “big sets” become perfectly ordinary sets in NF, unlike in ZF where they are too large to be sets and exist only as “proper classes.”
But this freedom comes with strange quirks. In ZF, Georg Cantor’s famous theorem says that every set has more subsets than elements — the power set is always strictly larger. In NF, that’s not provable in full generality. The standard proof breaks the stratification rules. It is provable, however, that the set of all sets that consist of double-curly-wrapped elements (like {{x}}) is strictly larger than the original set. So NF has a built-in “curly-bracket transformation” (called the T function) that sometimes changes a set’s size and sometimes doesn’t. Sets where |X| = |{{x} : x ∈ X}| are called Cantorian; if the mapping itself exists as a set, they are strongly Cantorian. Small finite sets are strongly Cantorian, but the set of natural numbers and the universal set misbehave. This T-function weirdness is, in a way, the old paradoxes bubbling up in a manageable form.
Another surprise: in NF, the Axiom of Choice fails. The Axiom of Choice says you can always pick one element from each set in a collection, even an infinite one. In NF, certain huge collections don’t admit such a choice function. But so far all the failures involve those big sets; the wellfounded sets might still obey full choice. So ordinary math about real numbers and calculus is safe.
Did NF Really Work? A Long Wait for a Proof
While ZF was quickly trusted to be consistent (or at least as safe as math can be), NF’s consistency was a mystery for over 80 years. Early on, mathematicians discovered that NF proves the existence of infinitely many sets — it cannot describe a finite universe — and that it refutes the full Axiom of Choice. But no one could prove that NF itself wasn’t secretly contradictory.
A variant, NFU, which allows atoms (objects that aren’t sets) and weakens the axiom of extensionality, was shown to be consistent by Ronald Jensen in 1969. Jensen built a model using techniques from combinatorics and nonstandard models of Zermelo theory. Later, Marcel Crabbé proved the consistency of certain weaker fragments of NF. But pure NF, without atoms, remained stubbornly open.
Then, as of 2024, Randall Holmes and Sky Wilshaw appear to have finally proved that NF is consistent. Their proof is enormously complex, a Fraenkel-Mostowski construction that was later verified by the computer proof assistant Lean. It seems that the everything-set can play by consistent rules after all. This discovery opens the door to taking NF seriously as a genuine alternative foundation for mathematics, not just a clever trick.
Why Does It Still Matter? Two Rulebooks for the Same Game
The fight over ZF and NF isn’t just a technical squabble among logicians. It challenges the idea that “set” has one obvious meaning. If both systems can avoid the paradoxes and both can serve as a foundation for most of mathematics, then the concept of a collection doesn’t force a single set of rules upon us. We get to choose which constraints to build into our foundation.
In your own life, you meet this kind of choice whenever you make up a game. You might decide that a certain powerful move is banned entirely (like ZF banning the universal set). Or you might allow it, but with restrictions on when and how it can be used (like NF’s stratification). Both games can be fair, fun, and free of instant-game-over disasters. The two approaches don’t so much contradict each other as chart different territories. ZF tells the story of sets built up from nothing; NF tells a story in which the whole is a part of itself and still works. Knowing that both stories have a happy ending — and that a dream from 1937 only recently got its proof — reminds us that even the most “basic” ideas still hold surprises.
Think about it
- If a “set of all thoughts” existed, would it contain itself? Why might that be a problem — and could you fix it by changing how you talk about thoughts?
- When you design a game, do you prefer to ban a tricky rule outright or to keep it with careful limits? Which strategy feels more honest to you, and why?
- Can you think of something in the real world — a list, a collection, a club — that seems to create a Russell-style contradiction? How do people usually deal with it?
