“And that wraps up our section on Month Theory,” The Chronology professor said to his class, “Let us now move on to the next chapter.” He could tell by the groans of his students that class was almost over. He checked his watch. He had plenty of time. “Now, now, class,” he chided, “Did you honestly hope I’d end class early? Have I ever done that?” “No!” Sam, the teacher’s pet, eagerly spoke up, while the rest lazily shook their heads. “Now, let us devote the last minute of class to studying eternities,” the professor said, “First of all, does anyone know what an eternity is?” Unsurprisingly, Jared jeered “Listening to your lectures!” The professor chuckled along with the class. “That is an example, yes,” he said, “Can you explain why that is?” Jared was dumbfounded. “W-what? I was, er, only joking, sir.” He stammered. Several students stifled giggles at his expense. “Well, I’m not,” The professor said, “So I ask again: does anyone know what an eternity is?” Sam spoke up once more. “Isn’t an eternity, like, forever?” That was the answer he expected to hear. “Sometimes, yes, in a certain sense,” he said, “But that’s not very rigorous. For a useful definition, we need something quantitative. We need to know how long an eternity is. So how long is it?” Sam began, “It’s in-” The professor cut him off. “Someone who isn’t Sam, perhaps?” Silence. “I know you know the answer,” he said, “I know you’re all thinking it. I just need someone to say it.” “Infinitely long?” he heard, from somewhere in the room. “Yes!” he said, “Exactly! An eternity is an interval of time that is infinitely long. See, that wasn’t so hard, was it?”
“A standard way to construct an eternity,” he lectured, “Is to suppose that an hourglass with 2 seconds of sand in the bottom is flipped. Then, for every second that elapses, an additional second of sand is added to the top. The duration of this hourglass is an eternity.” A few students nodded to indicate understanding. “Now suppose that such an hourglass, we’ll call it A, is flipped at noon. Now suppose that an identical hourglass, B, is flipped at 1 PM of the same day. Which hourglass has the longer duration?” Silence. He pointed to a student at random. “You,” he said, “What do you think?” She looked around nervously. “Me?” “Yes, you,” he said, “It’s OK if you say the wrong answer, just say something.” “Well, A, I guess,” she said. “And why’s that?” he asked. “Well, ‘cuz it’s an hour longer,” she said. “And what is infinity plus 1?” he asked. “I guess infinity,” she mumbled. “Precisely! So you see, no matter when the hourglass is flipped, the duration is the same eternity. In fact, even if it’s flipped infinitely long ago in the past, its duration is still the same. Two times infinity is still infinity.” “So what you’re saying is that all eternities are the same length?” “Not quite,” he said, “All eternities of this type, called countable eternities, are the same duration. But there is a second type of eternity, called an uncountable eternity, which is far longer. Can anyone think of what might be an example?” The entire class looked confused and bored in varying measure, except for Sam, who was visibly straining not to raise his hand. “Perhaps you can think of an example, Jared?” He said. Jared, who had sunk into his chair to avoid attracting attention, was startled upright. “Uh… your class?” “Very good!” The professor said, “Or a second or a minute or a year. Any finite interval of time contains an eternity which is longer than the previously discussed eternities.”
“But that doesn’t make sense,” Amelia interjected, “An interval can’t contain any intervals of larger duration, much less infinitely many of them. So how can infinity seconds be shorter than one second?” “That’s an excellent question,” the professor said, “in fact, you’ve just described the well-known ‘Eternity Paradox’, a famous unsolved problem. But a key part of it is measurement. A countable eternity is measured in seconds, which have duration, but an uncountable eternity is measured in moments, which have 0 duration. And the number of moments in a second is greater than the number of seconds in a countable eternity.” “So there are more than infinity moments in a second?” She asked, “Then how many are there?” “Well, it’s a bigger infinity,” he said, “you know how there are more irrational numbers than rational numbers? It’s kind of like that. But you’d have to ask a mathematician for a full explanation.” From the defiant look in her eyes, he could tell that she would. His work was done. “Well, that’s enough for today,” he said, “I don’t want to keep you over time, so class is dismissed…” He looked down at his watch. “Now!” The class all checked their phones, and began muttering in confusion and disbelief. “Huh? That was all in a minute?” “No way, he must’ve lied.” “Maybe it really was an eternity…” Their muttering died down as they all shuffled out into the hall.
All but one. Even after the rest of the students left, Sam still sat in his seat. “Hey, how come you didn’t tell them?” He asked, “Er, I mean, why didn’t you tell them, sir?” “Tell them what?” The professor asked. “How you’re the leading expert on eternities! How you practically invented them!” “It wasn’t relevant to the course,” the professor said, “besides, I’m nothing special. I just took some ideas from Cantor and Zeno and shuffled them around a bit.” “But no one else would have thought up a longer eternity that can fit inside a single second. How did you even come up with that idea?” “Well, son,” the professor said, “a long time ago – an eternity ago, in fact – your mother said that she would love me forever.”