Is Every Math Statement Either True or False?
The Mathematician Who Doubted “True or False”
In 1907 a Dutch mathematician named L.E.J. Brouwer (1881–1966) made a startling claim. Mathematics, he said, is not a collection of timeless truths waiting to be discovered. It is something we build ourselves, step by step, inside our minds.
If that sounds strange, think about what it means for something to be “true” in math. Most people learn that every statement is either true or false, and there’s no third option. This rule is called the law of excluded middle. “There is a biggest number” — false. “Seven is prime” — true. Even if nobody knows which yet, the law says the answer is already out there.
Brouwer said no. For a claim to count as true, you have to lay out a mental construction that proves it. If nobody has built such a proof, the statement isn’t automatically false either — it’s simply not settled. Imagine someone says, “There is a dragon hiding in your garage.” You wouldn’t say, “Well, either the dragon exists or it doesn’t — one of those must be true,” and then walk away satisfied. You’d want evidence. For Brouwer, mathematics works the same way.
This idea launched a movement called intuitionism, and it eventually forced mathematicians to rebuild even the most basic parts of their subject — including the theory of what sets are.
Building Sets Without a Magic Wand
Classical set theory, invented by Georg Cantor and formalized by Ernst Zermelo and Abraham Fraenkel, describes a universe of sets built from nothing. You start with the empty set — a box with nothing inside. Then you keep forming new sets from old ones. You can make a pair, a union, or the power set: the set of all possible subsets of a given set. For an infinite set, like the counting numbers, the power set is unimaginably huge. It contains every possible way of picking some numbers and leaving others out.
In the 1970s, the philosopher and mathematician John Myhill (1923–1987) asked: how can we honestly claim that “the set of all subsets of the natural numbers” exists? We can write down some subsets using rules — all evens, all primes, all numbers whose names contain the letter “e” — but we have no recipe for generating every arbitrary subset. There is no way to finish the list. Myhill wrote, “we have no idea of what an arbitrary subset of an infinite set is; there is no way of generating them all and so we have no way to form the set of all of them.”
So Myhill proposed a swap. Instead of the power set, use the exponentiation axiom: the set of all functions from one set to another. A function is a rule that takes each input to exactly one output — a recipe you can actually write down. You can form the set of all such rules without pretending you’ve already collected every possible mysterious subset. Myhill’s move let mathematicians carry out calculus and much more, while staying on firmer constructive ground.
The Danger of Circular Definitions
Why did the “all subsets” idea bother constructive thinkers so much? The answer involves a cautionary tale from the early 20th century. The philosopher Bertrand Russell (1872–1970) noticed that some definitions are circular in a harmful way. Imagine a barber who shaves everyone in the village who does not shave themselves. Does the barber shave himself? If he does, he shouldn’t; if he doesn’t, he should. The definition falls apart because it secretly refers back to the barber himself. Russell called the rule that bans such tricks the Vicious Circle Principle: a definition shouldn’t talk about the very thing it’s trying to define.
In set theory, an impredicative definition does exactly that — it defines a new set by referring to a totality that includes the set itself. For example, if you define a set of numbers as “all natural numbers n such that every subset of the natural numbers has some property involving n,” you’re secretly including your new set among those subsets before it’s even been built.
Classical set theory allows impredicative definitions freely. The power set axiom and the unrestricted separation schema (which lets you carve a subset out of an existing set using any property you like) both open the door to this kind of circularity. Constructive mathematicians put up a fence. In Constructive Zermelo‑Fraenkel set theory (CZF), separation is restricted: the property you use may only talk about sets you’ve already constructed, never about the whole universe. This keeps definitions cleanly bottom‑up, like stacking blocks one at a time without ever lifting yourself by your own bootlaces.
Two Systems: The Intuitionists and the Constructivists
Once mathematicians accepted that intuitionistic logic — logic without the law of excluded middle — might be a legitimate foundation, a new choice appeared. Which set‑theoretic principles should we keep?
One camp, led by Harvey Friedman (b. 1948), created Intuitionistic Zermelo‑Fraenkel set theory (IZF). IZF keeps almost all the classical axioms, including power set and full separation, but builds them on intuitionistic logic. The only classical rule they had to drop was the axiom of foundation, which forced an excluded‑middle‑style conflict. The result is an impredicative theory that is, in terms of proof strength, exactly as powerful as classical ZF. Its supporters say: give the mathematician the strongest tools possible, so long as the logic stays constructive.
A different camp, following Myhill and later the logician Peter Aczel (1941–2023), built CZF. This theory is doubly careful: it uses intuitionistic logic and tames the axioms themselves. Power set is replaced by subset collection, a principle weaker than power set but still strong enough for doing real analysis. Separation is restricted to bounded formulas. The result is a predicative theory — one that never defines a set by talking about the whole universe at once.
Aczel showed that CZF fits brilliantly with another approach: Martin‑Löf type theory, a system where every proof is a program and every set is built from trees. In that interpretation, the sets of CZF literally grow like branches on a tree — always unfinished, always open. This image matches the philosophical heart of constructive mathematics: the mathematical universe is not a finished museum we walk through, but a structure we are still building.
Why This Still Matters (Even If You Don’t Build Sets)
You might wonder why a debate about invisible sets matters to someone who just wants to use math, not rebuild it. The answer hides inside the devices you use every day.
When a computer program runs, it doesn’t appeal to the law of excluded middle. It can’t say “this loop will either stop or run forever, so I’ll just assume one is true.” It has to compute an answer, step by step. Modern programming languages and proof assistants are deeply influenced by constructive logic. A constructive proof that something exists often carries within it a recipe for actually finding it — which is exactly what you want if you’re writing software.
There is also a deeper human question. If you believe mathematical objects exist in a timeless realm, independent of us, then impredicative definitions and vast completed infinities feel natural — you’re just describing a landscape that was always there. But if you believe that we make mathematics, that it grows from our minds and our rules, then constructive set theory feels like honesty: you should only claim you’ve built what you can actually show.
Neither side has fully won. IZF and CZF both have passionate defenders. But the struggle they represent — between finished truth and open‑ended construction — appears every time you try to prove something new, or every time you wonder whether a puzzle even has a solution. The constructive mathematicians didn’t just change a few axioms. They reminded us that the most abstract question — “what can we truly say exists?” — might need to be answered one block at a time.
Think about it
- If a friend says, “There must be a way to win this game, even if nobody has found it yet,” are they relying on the law of excluded middle? Does that make their claim weaker?
- Imagine you could program a machine to list every possible subset of the natural numbers. Would that make the power set “constructed,” or does the very idea of listing all subsets contain a hidden shortcut?
- Is it more satisfying to think of mathematics as a huge map waiting to be discovered, or as a tower you get to help build? Why?
