Can maths prove everything?

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This is the episode's narration, word for word. Headings jump to that point in the video.

This sentence is false 0:00

Here's a puzzle. Read this sentence: this sentence is false. Is it true? If it's true, then what it says holds, so it's false. But if it's false, then what it says is right, so it's true. It flips forever and never lands. Hold on to that loop, because it unlocks a much bigger question. Can maths prove everything?

An old knot 0:27

That puzzle is the liar paradox, and Greek philosophers were already arguing about it more than two thousand years ago. But in 1930 a twenty-four-year-old Vienna logician, Kurt Gödel, turned a close cousin of it into one of the great results of mathematics. First, though, what is a proof?

Start from a few rules 0:49

Take Euclid, writing in Alexandria around 300 BC. His Elements begins with a handful of starting assumptions, called axioms. You can draw a straight line between any two points. All right angles are equal. From those, step by step, he proves things, starting with how to build a triangle with three equal sides. A proof is a chain where every link follows by the rules from the axioms, or from earlier links. Book one alone climbs from five postulates and five common notions to 48 results.

The awkward fifth 1:30

But Euclid's fifth postulate, about when two lines meet, was long and clunky, and for centuries people tried to prove it from the other four. Ptolemy and Proclus both thought they had, and both were wrong. In the 1820s, Bolyai and Lobachevsky built whole geometries where the fifth fails. In 1868, Beltrami showed the new geometry was just as consistent as Euclid's. So the other four can't settle the fifth, either way. Remember that: a statement the rules can't decide.

Proof by machine 2:10

By Gödel's time, mathematicians had tightened this into a formal system. You fix the symbols, the axioms, and rules so mechanical that a machine could check a proof without understanding a word. Then ask two sharp questions. Is it consistent, so it never proves a statement and also its opposite? And is it complete, so for every statement it can express, it proves either the statement or its opposite?

We must know 2:41

The German mathematician David Hilbert wanted both answers to be yes. In 1900 he set the world 23 problems, and the second was to prove that the axioms for the real numbers could never contradict each other. In the 1920s he launched a programme to write all of mathematics as a formal system, and prove, by simple reasoning, that it could never contradict itself. He also wanted a mechanical way to decide any question. In 1930, Königsberg, the city of episode nine's bridges, made him an honorary citizen, and he ended his speech: we must know, we will know.

A casual remark 3:23

That same year, in that same city, there was a famous conference. On the seventh of September, Gödel announced his result in what's been described as a casual remark, during a discussion. In the audience, John von Neumann understood at once how important it was. The paper came out in January 1931, promising formally undecidable statements.

Sentences into numbers 3:51

Gödel's first trick was to turn maths into numbers. Give every symbol a code number. Then use the codes as powers of the primes: two to the first code, times three to the second, times five to the third, and so on. With a toy code, nought equals nought becomes 2430. As in episode six, every whole number bigger than one splits into primes in only one way, so you can always decode it. A proof is a list of formulas, so it gets a number too.

Maths talking about itself 4:27

Now something strange happens. "This is a proof of that" becomes a fact about two numbers, checkable by ordinary arithmetic. So statements about whole numbers can, in code, talk about what the system proves. Then Gödel found a recipe for a sentence that, decoded, says: the formula with this number has no proof. And the number it names is its own. In effect, it says: I can't be proved in this system.

Nothing circular 4:59

Gödel saw the family resemblance himself, and wrote that there was a close relationship with the liar. But he insisted that, in spite of appearances, there is nothing circular about it. His sentence is a definite statement about whole numbers, in the system's own symbols, that just happens to describe itself.

Not false, unprovable 5:21

Here's why it's not a paradox. Swap false for unprovable. Suppose the system could prove the sentence. Then it would prove something false, because the sentence says it has no proof. A system that only proves true things can't do that, so the sentence has no proof. But that's exactly what it says. So it's true, and the system can't prove it. The liar loops forever; Gödel's sentence lands in a gap between true and provable in this system.

Exactly what it needs 5:55

That's the first incompleteness theorem, and it has fine print. The system must be consistent. There must be a mechanical test for whether something is an axiom. And it must handle basic whole-number arithmetic, adding and multiplying. Then some statement in its own language can be neither proved nor disproved. Gödel's proof leaned on a stronger assumption; in 1936 Barkley Rosser showed plain consistency was enough.

Can't we patch it? 6:27

So why not add the sentence as a new axiom? You can, but the new system has its own new unprovable sentence. Gödel's result covers any bigger system you build this way, as long as a machine can still check its axioms and it stays consistent. And each condition matters. Arithmetic with adding but no multiplying is complete. Make every true statement of arithmetic an axiom and you're complete too, but no machine could check your list.

The letter 7:00

Then came the second blow. On the 20th of November 1930, von Neumann wrote to Gödel about a remarkable consequence. Gödel had found it too, and it's in the 1931 paper: a consistent system like this can't prove its own consistency. Hilbert's programme could not be carried out as he had imagined it.

The second theorem, carefully 7:24

Carefully, then. If such a system is consistent, it can't prove the statement that it's consistent. That's less alarming than it sounds. A system that contradicts itself can prove anything at all, including its own consistency, so a proof from inside was never going to reassure anyone. And consistency can be proved from outside. In 1936 Gerhard Gentzen proved arithmetic consistent, using one principle arithmetic itself can't prove.

What it doesn't say 7:57

So what didn't Gödel prove? He didn't prove maths is broken. As the logician Torkel Franzén put it in 2006, mathematicians are by no means floundering in a sea of undecidability, and no famous problem about whole numbers had been shown undecidable in standard set theory. It adds nothing to claims about the law, the Bible or physics, which aren't formal systems. And it doesn't mean anything goes: it's always about one particular system.

Minds and machines 8:30

One more overclaim has famous champions. In 1961 the Oxford philosopher John Lucas argued that Gödel's theorem shows the mind can't be a machine, and Roger Penrose later made similar arguments. We can see the Gödel sentence is true, they said, and the machine can't. But you can only see that it's true if you already know the system is consistent, and the Stanford Encyclopedia of Philosophy reports a wide consensus that these arguments fail, though Lucas and Penrose have replied.

The machine that lists every proof 9:06

Machines bring us to Alan Turing. In 1935, as a young Cambridge mathematician, he heard lectures on Gödel and on Hilbert's dream of deciding everything. Turing noticed that if every statement could be proved or disproved, a machine could simply list every proof until it hit the statement or its opposite. In 1936 he showed no machine can decide, for every statement, whether it can be proved, and Alonzo Church reached the same answer independently.

Does it ever stop? 9:40

Today the cleanest version is called the halting problem, though Turing never used that name. Could a checker look at any program and say whether it will ever stop? Suppose it could. Build a contrary program that asks the checker about itself, then does the opposite. Told it stops, it loops forever. Told it loops, it stops. So no such checker can exist. It's the same kind of self-reference trick Gödel used, aimed at machines instead of proofs.

A real question nobody can settle 10:17

Are these just strange, made-up sentences? Back in episode two, Cantor showed there are more real numbers than whole numbers. Is there a size of infinity in between? Cantor guessed no, the continuum hypothesis, and it was first on Hilbert's 1900 list. In 1938 Gödel showed the standard axioms can't disprove it. In 1963 Paul Cohen showed they can't prove it either.

Independent, like the fifth 10:47

So the continuum hypothesis is independent of the standard axioms, if those axioms are consistent at all, just as Euclid's fifth was of the other four. One careful note: that came from Gödel's and Cohen's work on sets, not from the incompleteness theorem itself. Mathematicians still argue about it; some hope new axioms will settle it, others doubt there's a single answer. It's an open question, not a broken one.

So, can maths prove everything? 11:18

So, can maths prove everything? No single consistent, machine-checkable system strong enough for arithmetic can settle every question it can ask. Step up to a stronger one, and it has gaps of its own. This season we went from a little circle worth a hundred and eighty, through infinities, endless decimals, doors, birthdays, primes, pi, a tortoise and seven bridges, to the limits of proof itself. So what's the next small question that took centuries to crack? We'll find out together next season.

Sources

Every factual claim in the episode is tied to one of these. Spotted an error? Tell us.

1. Stanford Encyclopedia of Philosophy, "Liar Paradox" — plato.stanford.edu
2. Stanford Encyclopedia of Philosophy, "Gödel's Incompleteness Theorems" — plato.stanford.edu
3. Kurt Gödel, On Formally Undecidable Propositions of Principia Mathematica and Related… — archive.org
4. J. J. O'Connor and E. F. Robertson, "Kurt Gödel", MacTutor History of Mathematics,… — mathshistory.st-andrews.ac.uk
5. J. J. O'Connor and E. F. Robertson, "Euclid of Alexandria", MacTutor History of… — mathshistory.st-andrews.ac.uk
6. Euclid, Elements Book I (definitions, postulates, common notions, propositions), ed.… — mathcs.clarku.edu
7. J. J. O'Connor and E. F. Robertson, "Non-Euclidean geometry", MacTutor History of… — mathshistory.st-andrews.ac.uk
8. Stanford Encyclopedia of Philosophy, "Hilbert's Program" — plato.stanford.edu
9. J. J. O'Connor and E. F. Robertson, "David Hilbert", MacTutor History of Mathematics,… — mathshistory.st-andrews.ac.uk
10. Torkel Franzén, "The Popular Impact of Gödel's Incompleteness Theorem", Notices of the… — ams.org
11. Richard Swinburne, "John Randolph Lucas, 18 June 1929 – 5 April 2020", Biographical… — thebritishacademy.ac.uk
12. J. J. O'Connor and E. F. Robertson, "Alan Mathison Turing", MacTutor History of… — mathshistory.st-andrews.ac.uk
13. A. M. Turing, "On Computable Numbers, with an Application to the… — doi.org
14. Stanford Encyclopedia of Philosophy, "Turing Machines" — plato.stanford.edu
15. Stanford Encyclopedia of Philosophy, "The Continuum Hypothesis" — plato.stanford.edu
16. J. J. O'Connor and E. F. Robertson, "Paul Joseph Cohen", MacTutor History of… — mathshistory.st-andrews.ac.uk

Image credits

• David Hilbert, photo before 1912 · unknown photographer · public domain · via Wikimedia Commons
• Alan Turing aged 16, 1928–29 · possibly by Arthur Reginald Chaffin · Turing Archive, King's College Cambridge · public domain · via Wikimedia Commons

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