The Useful Unknowable

How Godel’s ghost is quietly rewriting cryptography, AI, and physics.

A graduate student just used the limits of mathematics itself to hide secrets. Across four fields this week, researchers are discovering that what we cannot know is becoming our most powerful resource.

In 1931, a twenty-five year old Austrian logician named Kurt Godel proved that any sufficiently rich system of mathematics contains true statements it can never prove. For most of the twentieth century that result was treated like a tombstone laid over a particular kind of dream, the dream of complete and tidy knowledge. Hilbert wanted everything decidable. Godel handed him a polite shrug and a counterexample. This past week, on a server quietly maintained by the International Association for Cryptologic Research, an MIT graduate student showed that Godel’s so called limit can be turned into a key. Once you start looking, the same trick is showing up in physics, in AI, and in the philosophy of mind, often in the same news cycle.

A Proof That Reveals Nothing (And Why That Is Hard)

Imagine, for a moment, that you have solved a fiendish Sudoku and you want me to believe you. The obvious move is to show me the grid. But suppose you would rather not. Suppose you want to convince me that the solution exists, that you possess it, that the puzzle is well posed, all without revealing a single digit. That is a zero knowledge proof. The idea was invented in the mid 1980s by Shafi Goldwasser, Silvio Micali, and Charles Rackoff, and for a long time it looked like a piece of elegant logical jewellery. Today it underpins billions of dollars of cryptographic infrastructure, from Zcash to private rollups on the new Ethereum stack to a small army of identity verification schemes.

There has, however, always been a catch, and the catch wears a name in cryptography circles. You can build a zero knowledge proof that is non interactive, meaning one message and no back and forth chatter. You can build one that is perfectly sound, meaning that literally no cheating proof exists in the mathematical universe. You can build one that requires no trusted setup, meaning there is no shadowy figure in a Geneva conference room running a one time ceremony to bake in some master parameters. You can have any two of these properties. The third, classical impossibility results said, is off the table. Pick two and move on.

Enter Rahul Ilango, an MIT PhD student whose paper Godel in Cryptography landed on the IACR ePrint archive last summer and which Quanta Magazine spotlighted on May 11. Ilango did not repeal the impossibility theorem, because impossibility theorems do not repeal. He side stepped it. He defined a fourth option which he calls effectively zero knowledge, and the trick is genuinely lovely. The escape route a would be cheater must take is engineered to depend on resolving a question that the standard axioms of mathematics, ZFC, can neither prove nor disprove. The lock is not unpickable in the absolute sense. It is unpickable in any universe whose mathematics obeys the mathematics we use to do mathematics. Which, as far as anyone has been able to tell since 1931, is the universe we live in.

The classical trilemma, and the corner Ilango invented to escape it

It is the sort of move that makes you laugh once you see it. The cryptographer is essentially saying that the cheater is welcome to break this protocol, provided they first settle a question Godel proved cannot be settled. Good luck. Send a postcard.

Godel’s Real Trick, For People Who Glazed Over in College

A quick refresher, because the rest of this only works if the foundation is in place. Skip the formal logic and just hold one sentence in mind. This sentence cannot be proved. If the sentence is false, then it can be proved, and so mathematics proves a false thing, which is worse than the original problem. Therefore the sentence must be true. Therefore there exists a true statement that mathematics, sitting inside its own walls, cannot reach. That is incompleteness in a nutshell.

Godel did not destroy mathematics. He bounded it. He showed that the price of being powerful enough to talk about ordinary arithmetic is having blind spots. Any formal system rich enough to be interesting is rich enough to have things it cannot see. The universe of mathematical truth, in other words, is bigger than the universe of mathematical proof. For ninety years the working consensus was that this was a melancholy fact, the kind of thing you noted in the introduction of a thesis and then carefully avoided in the rest of the document.

Ilango’s move is philosophically new in a way that is worth pausing on. He is, so far as I can tell, the first cryptographer to take a Godelian blind spot and use it. Not lament it, not work around it, but treat it as functional infrastructure. The unknowability is not a wound. It is a hiding place. And once you reframe a limit that way, you start noticing how often the same reframe is happening elsewhere.

The Black Boxes We Built, And Cannot Quite Open

Mechanistic interpretability is the unglamorous name for the wave of research currently treating large neural networks as alien organisms being autopsied feature by feature. The phrase MIT Technology Review picked up, alien autopsy, is not actually unfair. Sparse autoencoders pry open the hidden activations of large language models, identify recognisable concepts inside, and produce papers with titles like a child cataloguing strange creatures from a tide pool. Anthropic, DeepMind, and a growing band of academic labs are doing this in earnest, and the corpus is now thick enough that ICLR 2026 had whole sessions devoted to it.

The deep point, the one that does not get said often enough, is this. We trained these systems by gradient descent. Nobody wrote the algorithm they ended up running. There is no source code in the human sense, just weights, and weights are not legible the way Python is legible. The position we find ourselves in is to reverse engineer, with great difficulty, software we built by accident. A seventy billion parameter model contains structures we cannot, using any current technique, fully describe in compressed human form. The system is in some sense more powerful than our ability to formalise what it does, which is a sentence I would have considered melodramatic five years ago and which I now consider plain reportage.

On May 14, a joint group of researchers from Oxford, Google DeepMind, OpenAI, Anthropic, and Stanford posted a paper called Positive Alignment: Artificial Intelligence for Human Flourishing. Its move is the one we keep seeing. Instead of pre specifying every bad behaviour an AI must avoid, which is the dominant negative alignment paradigm of refusal training and content filters, the authors argue for training systems toward open ended human flourishing. The implicit admission is striking. The space of acceptable behaviour cannot be enumerated in advance. They are choosing to build with the unspecifiability rather than against it. Same shape as Ilango. Different room.

When You Cannot Look Directly, Look Sideways

Two stories from physics in the past two weeks carry the signature. Around May 1, the Oxford trapped ion group published a piece in Nature Physics demonstrating quadsqueezing, a fourth order quantum effect that classical squeezing techniques cannot reach. Their trick was non commutativity, the quantum world’s most famous obstruction. Combine two operations whose order matters and the gap between AB and BA becomes a resource. Their interaction is generated about a hundred times faster than conventional methods because the order of operations gap is itself the amplifier.

Two paths, two states, one difference. The gap that Heisenberg taught us to respect is now also taught to do work

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