The Sentence That Says It Can't Be Proven

The cafeteria-turned-competition-room had emptied an hour ago. Folding chairs leaned against the wall in crooked stacks. On the whiteboard, the last tiebreaker problem was still up there, unsolved by anyone, including the team that won.

"It's not even hard," Maya said. She was sitting on a desk with her shoes off. "It just doesn't end."

Soren had his notebook open on his knees. "Read it again."

Maya read it. "Write a statement about numbers that is true but cannot be proven true."

"That's not a real problem," Soren said. "That's a trick. If it's true, you prove it's true, and then it's proven."

"That's what everybody said. That's why they all left." Maya swung her feet. "But the judge wrote it. A judge wouldn't put a fake problem as the tiebreaker."

"Maybe she made a mistake."

"Maybe." Maya didn't sound like she believed it. "Try one."

Soren wrote a sentence. "This statement is false."

Maya read it upside down. "That's not about numbers. And it's not true or false, it just spins. If it's false then it's true then it's false."

"I know," Soren said. "I wanted to see it spin." He crossed it out. "The problem says true. Spinning isn't true."

Maya slid off the desk and walked to the board. She uncapped a marker and didn't write anything. "What if we make it talk about being proven instead of being false. Watch." She wrote: This statement cannot be proven.

Soren went quiet, reading it. Then he said, slowly, "Okay. Pretend you prove it."

"Pretend I prove it."

"Then it's been proven. But it says it can't be proven. So if you prove it, you've made it false. You proved a false thing."

Maya tapped the marker against the board. "Proofs aren't allowed to do that. A proof of a false thing is broken."

"Right. So you can't prove it. The system isn't allowed to prove it." Soren was writing fast now. "You can't prove it without breaking."

"So it can't be proven," Maya said.

"So it can't be proven."

They both stopped.

"Wait," Maya said. "Say what it says again."

Soren read it off the board. "This statement cannot be proven."

"And we just figured out it can't be proven."

"Yeah."

"So it's telling the truth."

Soren looked up from the notebook. "It's true."

"It's true," Maya said, "and we can't prove it."

For a second neither of them said anything. The radiator ticked. Maya put the cap back on the marker and didn't move.

"That's the answer," Soren said. "That's the actual answer. The problem wanted us to find a sentence that's true and unprovable, and the sentence about being unprovable is the sentence."

"We didn't prove it though." Maya turned around. "We just sort of, walked around it. We can see it's true from outside. From over here. But the math itself, the proof-machine, it can never get there."

"Because the second it touches it, it breaks."

"Right."

Soren stood up, holding the notebook against his chest. "That means there's stuff that's true that you can never, ever reach. Not because you're not smart. Not because nobody's tried hard enough. Because it's actually not reachable. There's a wall, and the wall is built into the rules."

Maya was already shaking her head, but not in disagreement. "It's not one sentence, Soren. We just made one in like ten minutes. You could make a million. You could make one for every rulebook."

"Every rulebook," Soren repeated.

"Any set of rules strong enough to do real math. You could always, always write a sentence it can't reach." She looked at the board. "You could add a new rule to catch this one. Make a bigger rulebook."

"And then," Soren said, getting there with her, "the bigger rulebook has its own sentence it can't reach."

"And you make a bigger one."

"And that one has one too."

"Forever," Maya said. "You can never finish. There is no rulebook that gets everything."

Soren sat back down, hard, like his legs had decided for him. He opened the notebook to a clean page and his pen started moving, the letters small and fast and crowded.

Maya came and sat next to him on the floor with her back against the desk leg. "This was somebody's real thing, wasn't it. Not just a tiebreaker."

"It feels too big to be just a tiebreaker."

"Somebody figured this out for real. A long time ago." Maya pulled her knees up. "And they must have hated it. Everybody wanted math to be the one thing that's totally finished. Where if something's true you can always show it. And then somebody proved you can't. They proved that you can't prove everything."

Soren stopped writing. "They proved a thing about what can't be proven."

"Yeah."

"That's the part that gets me." He stared at the board. "They could prove the wall is there. They just can't get over it."

Maya was quiet. Then she said, "I like that there's a wall."

Soren looked at her.

"My whole life people act like everything's already figured out and I'm just slow to catch up. Like all the answers are in some book and I should stop asking." She picked at the carpet. "But there isn't a book. There can't be a book. There's stuff that's true forever that nobody will ever finish reaching. That means the asking doesn't run out."

Soren almost smiled. "The asking can't run out. It's a theorem."

"It's a theorem," Maya said.

The overhead lights buzzed. The sentence sat on the whiteboard in Maya's blocky letters: This statement cannot be proven. True, and standing there, and unreachable, like a star you can see but never walk to.

Footsteps came down the hall. The judge leaned in the doorway, jacket on, keys in her hand. "You two are still here. I'm locking up. Did you crack it?"

Maya and Soren looked at each other.

"No," Maya said. "Nobody can."

The judge smiled like that was the right answer, and reached over, and turned off the lights. In the dark the whiteboard still held its faint glow, the sentence pale gray on the gray, the marker uncapped on the tray where Maya had left it.