The List That Wouldn't Finish

The rain had been going so long that Soren's mom stopped commenting on it. On the fridge, two magnets sat next to each other. A one. A two.

"Bet you can't count all the numbers between them," Maya said.

"That's not a bet. That's just infinity."

"Prove it's infinity, then."

Soren pulled a sheet of paper toward himself and wrote a line of numbers. One point one. One point two. All the way to one point nine.

"See? Nine so far. But between one point one and one point two there's one point one one, one point one two." He kept writing. The paper filled. "It never stops. There's always a smaller step."

"Okay," said Maya. "So it's infinity. Same as counting one, two, three, four forever."

"Same infinity," Soren agreed.

Maya frowned. She did not say anything for a moment. Then she picked up the pencil.

"Line them up," she said. "If it's the same, we can match them. One whole number for one decimal. First decimal, first whole number. Second, second. Like giving everyone a seat."

"Fine." Soren drew a tall list down the left side of the page, numbered one to ten, room for more. "Put any decimals you want in the seats. Every single one between one and two. If they all fit, they're the same infinity."

Maya started filling seats. Seat one, one point three zero zero zero. Seat two, one point four one four two. Seat three, one point five nine two six. She wrote fast, whatever came, digits trailing off into more digits.

"Now I'm going to break your list," she said.

"You can't. There are infinite seats."

"Watch." She drew a small box around the first digit after the point in seat one. A three. Then the second digit in seat two. A four. Then the third digit in seat three. A two. A little diagonal staircase down through the list.

"What are you doing," said Soren, but he was already leaning in.

"Making a new number. One point something. And I pick each digit so it disagrees with the diagonal." She wrote it under her breath. "Seat one's first digit is three, so mine is four. Seat two's second digit is four, so mine is five. Seat three's third digit is two, so mine is three."

Soren stopped leaning and went completely quiet. His hand moved to his notebook without his eyes leaving the page.

"Keep going," he said.

"One point four five three, and it keeps going forever, one digit off from every seat." Maya tapped the new number. "Now. Which seat is it in?"

Soren checked seat one. "Can't be seat one. You made the first digit different."

"Right."

"Can't be seat two. Second digit's different." He ran his finger down the diagonal. "Can't be seat a hundred. Can't be seat a million. Wherever it would sit, you already made it wrong in that spot."

"So it's a real number between one and two." Maya sat back. "And it's not on your list. Anywhere."

"Make the list longer," Soren said quickly. "Add it at the bottom."

"There is no bottom, it's infinite. But sure, cram it in somewhere. Then I do the diagonal again and build another number that's missing." She spread her hands. "I can always build one you left out. Always."

Soren wrote it down in the notebook, slowly, the way he wrote things he wanted to still be true tomorrow. The whole numbers, he wrote. One seat each. And you can still make a decimal that doesn't get a seat.

"So they're not the same," he said. "The decimals are more. More than infinity."

"More than that infinity." Maya was looking at the fridge now, at the gap between the one and the two, a gap the width of two fingers. "There's a bigger infinity crammed into that little space than there is in counting forever."

Soren felt the kitchen get strange. The counting numbers had always been his idea of everything, the biggest thing there was, one two three going on past every star. And here was Maya's diagonal, quietly stepping past all of them.

"Wait," he said. "If there's a bigger one. Is there a bigger one than the bigger one?"

Maya's eyes went sharp. "Do the trick again?"

"Do the trick on the big infinity. Whatever it is. Make a list of those, do the diagonal, get something not on the list." He was writing and talking at the same time now. "You'd get something even bigger. And then you could do it to that."

"Forever," Maya said.

"Infinities all the way up." Soren put his pencil down. "Each one too big to fit in the one before it."

They sat there. Outside, the rain kept doing the one thing rain does. On the paper between them was a list that could never be finished and a number that would always be missing from it, and above that number, they both knew now, was another kind of endlessness, and above that another, a staircase with no top step.

"A man figured this out," Maya said. "He had to have. Somebody did the diagonal first."

"People said he was wrong," Soren said. He didn't know that for certain, but it felt right, the way a thing feels right when you've just done it yourself and can imagine everyone insisting it couldn't be done. "They probably said, you can't have more than infinity, that's what infinity means."

"And he had this." Maya put her finger on the little diagonal staircase, the boxed digits climbing down the page. "Just this. On a piece of paper. And it wins."

Soren looked at his own list of empty seats, all of them filled, all of them still not enough. He had spent his whole life sure that if he was just patient enough, careful enough, he could reach the end of a counting. That there was one everything and it belonged to the people who kept going.

"There's no biggest," he said out loud, and his voice came out smaller than he meant it to.

Maya reached over and drew one more box, on a digit in a seat they hadn't even used yet, starting the staircase again.

"Come on," she said. "Let's make one it doesn't have."

Soren picked up the pencil and began to write the number that wasn't there.