More Than All

The card said: A hotel has infinity rooms, all full. A new guest arrives. Where do you put them?

Soren read it three times. His cousin Bea had left a shoebox of these cards before she went back to university, and the closet under the stairs was the only quiet place left in the house with the rain hammering and his little brothers loose in every room.

He wrote in his notebook: move everyone up one room. Room one to room two, room two to room three. Forever. Room one is empty now. New guest goes there.

He sat back, pleased. The hotel was full and it still had room. That should have been impossible, and it wasn't, and he could feel his head stretching to hold it.

Next card: Now infinity new guests arrive, a whole busload, infinite passengers. Fit them all.

That was harder. He chewed the pen. Move everyone to double their room number. Room one to room two, room two to room four, three to six. Now every odd room is empty. Infinite odd rooms. Infinite guests. He matched them up, one passenger per odd room, and none left over on either side.

He was grinning at the wall by then. You could add infinity to infinity and it fit. You could pour the bus into the full hotel and it swallowed the whole thing.

He started to think infinity was just a word for there is always one more. That whenever you needed room, you shoved everybody down the hall and made room. A trick that always worked.

The last card in the box was different. Bea's handwriting got smaller, like she'd been excited.

New guests arrive. Each one's name is an endless string of zeros and ones, going on forever. 0110101000... Every possible such string shows up, one guest each. Fit THEM in the hotel.

Easy, Soren thought. Same trick. He drew the rooms in a column and started writing a name beside each one. Room one, some string. Room two, another. Room three, another. He'd just list them all and match each guest to a room, like the bus.

He got three rooms down and stopped.

There were too many names. Not too many to write. Too many in a way he couldn't point at yet. He had the feeling first, the way you feel a step missing in the dark before your foot finds the empty air.

He stared at his list of three names. Then he did something Bea hadn't told him to do. He decided to build a guest who wasn't on the list. Any list. Even an infinite one.

Take the first digit of room one's name, and flip it. If it's a zero, write a one. Take the second digit of room two's name, flip it. The third digit of room three's name, flip it. Down the diagonal, forever, flipping every digit he touched.

He wrote the new name out, digit by digit, from his three rooms and then imagining the rest going on down the endless hall.

Then he checked it against the list.

It couldn't be in room one. He'd made its first digit different from room one's first digit. It couldn't be in room two. Its second digit was wrong for room two. It couldn't be in room three, or room four, or room nine hundred, or any room at all, because at that room's own digit he had flipped it. The new name disagreed with every single guest in at least one place.

Soren put the pen down.

He could do this no matter how the rooms were filled. Hand him any list of names, infinite as you like, going down the hall forever, and he could walk the diagonal and build one more name that wasn't anywhere on it. And then build another. The names didn't fit in the rooms. They never would. Not because the hotel was small. The hotel had infinity rooms.

There were just more names than there were rooms.

More than infinity. Not bigger numbers. A bigger infinity, sitting on top of the one he'd been so proud of, the way the second floor sits on the first.

He felt the closet get smaller around him and the world get bigger past the walls. The rain didn't sound like rain anymore. It sounded like every drop having its own room and there still not being enough rooms.

He thought, if there are two sizes of infinity, then.

He didn't finish the sentence on paper. He finished it in his chest. If there are two, why would it stop at two. He could take this new, bigger infinity, the one too big for the hotel, and ask the same question about it. Build a hotel out of those. Then find something too big for that hotel. And again. A floor above every floor. No top one. Ever.

He was eleven years old in a coat closet and he had just felt the ceiling of counting lift off and keep going up, floor after floor after floor, with no roof anywhere, not at the end, because there was no end.

His brother yanked the closet door open. Light fell in.

"Mom says dinner. What are you doing in here."

"There's more than all of them," Soren said.

"More than all of what?"

"All of them. Whatever you've got. There's more."

His brother stared at him, decided he was being weird, and left the door open.

Soren looked back at his notebook. The list of three names. The flipped name in the margin that belonged to no room. And Bea, somewhere across the rain, had known. She'd written this card with her handwriting getting small and quick because she'd known what it did to a person.

He found a fresh line and copied the diagonal name out clean, the one too big to fit, and then under it he wrote the first three flipped digits of a name too big to fit in the hotel made of THOSE, and his hand was not quite steady when he made the second one.