The List That Could Not Be Finished

The bus had not moved in eleven minutes. Maya had counted the minutes on the ticking sign, and then she got bored of minutes and started on windows.

"That tower," she said, pointing across the stopped traffic. "How many lit windows."

Soren looked. The apartment building went up higher than the bus window let him see. "You can't count them all. It goes past the top of the glass."

"Doesn't matter. Say it goes up forever. I could still count them. One, two, three, floor after floor. I'd never finish, but every window would get a number eventually."

"Sure," said Soren. "That's the whole numbers. One, two, three, no end. You'd hit every window if you had forever."

Maya liked that. A thing you could never finish but never miss anything in, either. "So that's infinity. The windows and the numbers are the same size of forever."

"Same size," Soren agreed. He pulled his notebook out and drew two columns, windows on the left, numbers on the right, a little line pairing each one. "You match them up. One window, one number. Nobody left out on either side. Same infinity."

The bus lurched, stopped again. Somebody groaned.

"Okay," said Maya. "Now do the numbers between the windows."

"What?"

"Between window one and window two. All the in-between numbers. One point one. One point five. One point nine nine nine. There's a whole crowd of them squished in there." She was already leaning forward. "Count those."

Soren started to say it was easy and then stopped, because he had actually started trying. "Which one comes first. After one point zero."

"One point one?"

"No, because one point zero one is smaller. It comes before that."

"One point zero zero one, then."

"There's always a smaller one. You can always jam another zero in." He tapped the pen. "So I can't even find the second one to count. There's no next."

Maya went quiet. Then, "But maybe you're just counting them in a dumb order. Maybe there's a clever order. A list. Where every in-between number shows up somewhere."

"Fine," said Soren. "Let's say I did it. Say I'm a genius and I made the perfect list. Every single in-between number, first, second, third, all the way down. Whole numbers on the left, in-between numbers on the right, all matched." He wrote a fake list, scribbling made-up decimals down the page.

0.1834... 0.5501... 0.2299... 0.7160...

"Every number in the world is somewhere on that list," he said. "Supposedly."

Maya stared at the scribbles. The bus sign ticked over another minute. And then something moved in her, the way it did when a puzzle turned a corner she hadn't seen was there.

"Make me a new number," she said slowly. "Off your list. I'll build it so it can't be on there."

"You can't. Every number's already on it."

"Watch. Take the first digit of your first number." She pointed. "One. I'll make my number's first digit not one. Say two." She grabbed the pen. "Second digit of your second number. Five. My second digit is not five. Say six. Third number, third digit, nine, mine's not nine, say zero."

Soren went still, watching her hand go down the diagonal of his list, one digit from each row, changing every one.

"So my number is zero point two six zero, and on and on," said Maya. "Now. Is my number the first one on your list?"

"No," Soren said, very slowly. "Its first digit is different from your first number's first digit. You made sure."

"Is it the second one?"

"No. Different second digit."

"The hundredth?"

"No, because you changed the hundredth digit off the hundredth number." He was leaning over the notebook now, and his voice had gone thin. "It's different from every single number on the list in at least one place. It can't be any of them."

"But it's a real in-between number," said Maya. "It's sitting right there between window one and window two. And your perfect list doesn't have it."

Soren looked up at her. "Do it again with a fixed list. Add my number to the bottom, make a new perfect list."

"Then I just do the diagonal again. I'll always build one you missed. Always."

The bus started moving. Neither of them looked out the window.

"So the windows lose," Soren said. He said it like it wasn't quite allowed. "The whole numbers go on forever, one two three, and it's not enough. There's not enough of them to number the in-between numbers. Even with forever."

"There's a forever," Maya said, "that's bigger than the counting forever."

"Bigger than infinity."

"Bigger than that infinity." She said it carefully, like stepping on ice to see if it held. "Which means there isn't one infinity. There's a small one and a big one."

Soren was already writing, the pen digging in. "And if there's a small one and a big one," he said, not looking up, "what stops it. What stops somebody doing the diagonal trick on the big one and finding an even bigger forever above it."

Maya opened her mouth. Closed it.

"Nothing stops it," she said. "You could keep going. Infinities stacked on infinities. A tower of them."

"Going up," said Soren, "past the top of the glass."

They both looked at that, at the idea of it, the way you look up at something too tall to see the end of. Maya felt the back of her neck go cold and bright at the same time. She had thought infinity was the last room in the building, the top floor, the place where counting finally stopped mattering because everything was in there.

It wasn't the top floor. It was the first one.

"There are people," Soren said, and his voice was almost a whisper, "whose whole job is climbing that. Right now. Tonight."

"How high have they gotten?"

"I don't know. I don't think it ends."

The bus pulled level with the tower. Up close the lit windows slid past faster than Maya could name them, row over row over row, and above the roofline the dark kept going, and she pressed her face to the cold glass and tried to count the floors she couldn't see.