Christian Kiefer - The Infinite Tides

He was only a few steps away from his goal. This was what he had been focused on when Quinn’s school year started and it was what he was still focused on when that school year ended and another began. He was so close to becoming an astronaut that it was all he could think of. Perhaps he had assumed that the vector upon which he had imagined his daughter to move continued to guide her progress. Or perhaps he had simply lost track of her. By then he might have lost track of both of them. He had worked so hard, so very hard, every day, for so many hours, and then the phone call from the Astronaut Office came, telling him that he had been accepted into the training program at long last. He was almost there. He went home and told Barb the news and she squealed in delight and leapt into his arms.

She seemed excited but later that same day, before Quinn came home from school, she told him that she did not think it was necessary that they all move to Houston with him, that he should go ahead and start the training and they would talk about moving her and Quinn later, after he had finished the training and was working regular hours at Johnson Space Center. “You’re going to be busy all the time, Keith,” she had said. “And we’ll just be a distraction anyway. You can come home when you can. On weekends or whenever you have a break. And Quinn’s just settled into her new school. Remember what happened when we moved last time? That’s hard on a kid like her.”

He tried to raise counterarguments but she shot them down one after another and there was that central important fact: that he was going to be busy, very busy, all the time. Still, he could not understand why she would want him to start his training without them. But maybe that was not it at all. Maybe she was only stepping aside so he could embrace his training more fully, so that he could charge, unencumbered, toward his destiny, toward their destiny, for they had chosen it together and now it was almost upon them. Maybe she was right and it was premature to move the entire family to accommodate the training period. They could be a family again in Houston when he was done with training. They had their entire lives, after all, and the best part had nearly arrived at last.

“OK,” he said, simply and definitively. “OK, we won’t move.”

Of course he had thought then that Quinn would be relieved for the same reasons that Barb had raised, for indeed he did remember her dramatic reaction to their last move: the weeks of sulking, the angry slamming of doors. Quinn was thirteen years old now and she was already doing the kind of work he had not even known was possible until he was eighteen or nineteen. They had a gifted program at the junior high, one significantly more advanced than the program at her previous school, and the level of mathematics to which she was being introduced was staggering and exciting. He still brought up the academy sometimes, continuing to hope that she might enroll there for high school, but she did not seem any more enthusiastic about the idea. And yet he could not help but think how much greater her experience at the academy would be nor could he understand why anyone would choose something lesser whan a greater solution presented itself, especially because her enthusiasm for mathematics had only grown in intensity. Sometimes she would call him to the kitchen table — still where she did most of her homework — and would ask him to doublecheck her numbers or would tell him about some project she was doing about black holes or perpetual motion machines or something else and he could feel her excitement, her discovery, the path of her forward motion.

The evening he was to tell her his good news, she was working on a math paradox called Hilbert’s Hotel, a puzzle he had forgotten about entirely concerning a hotel with an infinite number of rooms, all of which were full, and the various guests that arrive looking for a vacancy. She told him this with a smile on her face, as if imparting some impossible wisdom to her father, as if finally she had something to tell him that he did not know and he played along with that notion.

“So what happens then when an infinity of guests arrive and all of them are looking for a room?”

“I don’t know,” he said. “They tell them the hotel is full?”

“No, Dad, it’s an infinite hotel. Remember?”

“Right,” he said.

“Pay attention.”

“I am.”

“Infinite hotel. Infinitely full. Infinite people arrive and they all want rooms. What do they do?”

“I don’t know.”

She paused as if for dramatic effect. Then she said, “They ask every other guest to move down one room.” She was really smiling now. Beaming. In his memory it was like there was light shining from her. From her face. From all parts of her at once.

“How so?” he said.

“Look,” she said. She took a scrap of paper, already mostly covered with numbers, and wrote n on it and then said, “If n is a room with a guest then n moves to n plus one and then—” her pencil moving as she spoke, finishing the line and then sliding the paper around so it faced him. “There,” she said.

He looked at it, at the scribbles of numbers and lines and equations rambling across the page. “Nice,” he said. “That’s interesting.”

“That’s awesome.”

“That’s what I meant. Awesome. What’s that called again?”

“Hilbert’s Hotel. I can’t believe you haven’t heard of that. It’s kinda famous.”

“Yeah, maybe I have. Seems familiar. I probably forgot.”

“Oh yeah, you forgot. I doubt it.”

He smiled at her. “So do I,” he said. He leaned over the paper and took the pencil from her and wrote two symbols:

ℵ a

“What’s that?” she asked.

“Aleph,” he said. “It’s the symbol for the cardinality of infinite sets.”

“The whatsit?”

“The cardinality of infinite sets.”

She stared at him.

“The aleph symbol is the cardinality. It just means infinity here.”

“Infinity like the hotel?”

“Exactly like the hotel.”

“So then the little a is the set?”

“Correct.”

“Cool.”

“So how would you apply that to the hotel?”

She sat for a moment. Then she said, “Well, the hotel at start would be aleph- a .”

“Maybe,” he said. “Most mathematicians would probably call it aleph-null.”

“Null why?”

“Because that’s the smallest possible set.”

“OK, so the empty hotel would be aleph-null.”

“Yeah, OK,” he said.

“Then the infinitely full hotel would be aleph- a .”

“Aleph-one. Remember, the sets would be numbered. The a is just the variable.”

“Right right,” she said. “So aleph-one.”

“OK,” he said. “Then one more guest arrives and that would be what?”

“Aleph-two, I guess,” she said.

“I guess so,” he said. “So what does that mean?”

“I don’t know.”

“Think about it.”

She did and then said, tentatively, quietly: “Some infinities are smaller than other infinities?”

“Or larger,” he said.

“Or larger,” she said. And then, after a moment, she said, “Holy crap!” He smiled at her. “Yeah, holy crap,” he said.

“That’s awesome.”

“Well, yeah, kind of.” He wondered if he could get the same response if he taught her the mathematics of building something, how the numbers could predict the size or density or dimensions a piece of metal would need to be, how the numbers could be made into something solid and useful and tangible. Maybe he would try that next time. “You can’t do anything with it but it’s fun to think about,” he said.