Chapter 49 — Measure Zero
Rufus explores the concept of infinity.
Followed by Chapter 49 —— Measure Zero, in which Mica wonders if there might be other time travelers.
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Hello Friends,
It’s been a little while since we talked some mathematics, but today’s chapter touches on small infinite collections, as it were. So, let’s take a little side track.
Measure zero is in some sense a way of describing how one set, though being potentially infinite in size, can be completely inconsequential when compared to another infinite set. That is, it rests on the idea that there are different types of infinity. So, I thought it might be fun to explain the first two types of infinity that one typically encounters when starting out on their mathematical journey.
The first infinity that one typically encounters is so-called countable infinity. A countably infinite set is a collection of things that, though infinite, can nonetheless be counted. The quintessential example being the collection of natural numbers themselves: 1, 2, 3, 4, and so on, forever. This collection of numbers is obviously not finite; you could never make a list of every such number that you could run your eyes over. On the other hand, every number in this set has its rightful place in the list; it can be counted off.
Lest you might think I’m being a little obtuse in what I’ve just said, let me note that the collection of all integers, that is all whole numbers, including zero and the negative numbers as well, is a set of the same size. And, I hear you, that makes no sense, the natural (or counting) numbers are obviously a subset of all integers. Indeed, in some instinctive way, there are half as many of them … except that I can count off the set of all integers as well. Here’s one way to list them: 0, 1, -1, 2, -2, 3, -3, and so on. With this counting zero is the first number, 1 is the second number, -1 is the third number, 2 is the fourth number, -2 is the fifth number and so on. That is, I’ve given you a one-to-one correspondence between the collection of all integers and the natural numbers, with every number in each set accounted for exactly once. I mean, if I can put the sets in one-to-one correspondence, they must be the same size, right? Weird, huh?
Buckle up, though, because it’s about to get weirder: the collection of all rational numbers——numbers that are a ratio of two natural numbers, aka fractional numbers——is also countable. Yes, that means throw in all the sevenths and 123’rds and every other possible fraction and you don’t get a bigger set. For those among you who like a challenge, pause here and see if you can line up a correspondence between every rational number and the natural numbers alone. As a hint, consider lining up all fractions with “1” in the denominator (in other words the integers), and then under that line, do the same with all fractions with a “2” in the denominator. And beneath that, a line with “3”’s in the denominator. Continue doing this and then try starting at the top left corner (at 1), snake across one place to the right (to 2), and then diagonally down and left (to 1/2), then go down one place (to 1/3), before snaking up and right (skipping over 2/2, since we’ve already accounted for that when we saw 1/1) until you reach the first row again (3, this time). Now, head right one place before snaking back down and left until you reach the left column again. Keep up this back and forth action to make a chain of numbers (skipping over duplicates such as the “2/2” that we already skipped over). You now have every fraction listed off in a sequence. So, the set of rational numbers is countable!
As an aside, you could eliminate all duplicates that represent the number 1 by continuing a ray from 1/1 through 2/2 and beyond, simply striking out any other fractions you encounter (3/3 would be the next). Better still, for those among you who like pictures (and what mathematician doesn’t?), you could, in the same way eliminate all other duplicates by cleverly starting rays at “0/0” and striking out all but the first fraction that each ray hits. Can you see why this would work?
Anyway, at this point you might be thinking to yourself: sure, all infinities are the same, right? Infinity is infinity. Well hold on to your hat, now comes the truly mind-blowing piece. Suppose instead you considered every number between zero and one, that is all possible decimal expansions. For instance pi/4 is a number you didn’t already have, but now do. This set of so-called “real numbers” between zero on one is, I assert, a strictly bigger set of numbers than the counting numbers, even if the only counting numbers in this set are zero and one. And, because we can’t count this new collection of numbers, we call it uncountable in size.
But how can I be so sure about the size difference between the two collections of numbers? Well, I can explain it to you using a delightful proof by contradiction——that’s the mathematical term for an argument that supposes one thing is true, follows one’s nose to discover a contradiction, and, in doing so, shows that our original assumption must be false.
Let’s give it a go. Suppose, for the sake of this argument, that the real numbers between zero and one are countable; that it’s possible to make a list of them that includes every real number between zero and one. Consider then the first number on your list: it starts zero point something. If the first digit after the decimal place is a 2 then I’m going to start constructing a new number starting with 0.1, and if it’s not a 2 then I’m going to start with 0.2. Now, let’s look at the second digit of the second number on our list: if it’s 2, then I’m going to pick the second digit of the number I’m constructing to be 1, and if it’s not 2 then I’m going to pick my second digit to be 2. Perhaps you’re seeing the pattern, I can keep picking mismatched digits all the way down the line, and the number that I’m describing will not be anywhere on our putative list, which gives us a big Uh-oh! Actually, that’s exactly the contradiction we were looking for, meaning that since there is no list that doesn’t miss a number, there was no way to possibly count off every decimal expansion between zero and one. So now you’ve encountered your first uncountably infinite set too.
By the way, although we strictly only constructed one number not on the list, you can see by the way that we constructed it how easy it would be to generate many others.
Anyway, since I’m sure there are some of you out there for whom this is the first time you’ve considered different sizes of infinity, and that for those of you, your mind is probably melting right now, out of deference to those among us, let’s turn back to fiction and time travel in particular.
Just one last note: as I mentioned in last week’s commentary, next week’s chapter will include some images, so for those of you who are not already signed up for the email newsletter version this podcast, now’s a good time to consider doing so. It’ll deliver a text version of each episode, replete with images, right to your inbox every week.
Until next week, be kind to someone and keep an eye out for the ripples of joy you’ve seeded.
Cheerio
Rufus
PS. If you think of someone who might enjoy joining us on this experiment, please forward them this email. And if you are one of those someone’s and you’d like to read more
And now, without further ado, here’s chapter forty nine, in which Mica wonders if there might be other time travelers.
— 49 —
Measure Zero
Waiting for runway clearance, Mica gazed past Saskia, who was seated by the window. She took in the reflection of their plane in the panes of glass that encased the terminal building. No doubt there were passengers on the other side of the mirrored wall gazing right back at her, neither able to see the other, but both assuredly aware of their counterparts’ existence. “Given time travel exists,” she mused, “it’s weird that we haven’t met any other time travelers.”
Saskia turned to Mica and cocked her head. “How do you know we haven’t?”
“I think we’d know, if there were other time travelers.”
To Saskia, Mica’s contention was not nearly so obvious. Nobody else thought it of her. And even if everyone that they encountered considered the concept crazy, that didn’t mean that the timeline they were living in was a unique aberration; it was possible that there were other timelines and that in those time travel was commonplace.
A mathematical metaphor sprang to mind: the baron scarcity of the rational numbers on the number line, despite their infinitude. “You probably call them fractions,” she clarified. “Numbers made by dividing one whole number by another. A half. A quarter. Two thirds. You get the idea. Point is that even though there are infinitely many of them——in the same way that there might be many others who can slip in time——if you threw a dart at the number line, your odds of hitting one would be zero. Maybe most time travelers live in a parallel universe; there are tons of them by one measure, and almost none, by another measure. Literally measure zero, if I was being technical.”
Mica looked thoroughly confused. “What other numbers are there?”
“Well, square roots to start with. And cube roots etc. etc.”
“There are enough of those that I’d literally have zero chance of hitting a fraction?” Mica glanced around at the other passengers, but nobody was paying them any attention.
Saskia smiled. “Actually, even if we included every algebraic number——numbers that are solutions of a polynomial; equations with squareds and cubes etcetera in them——even all the algebraic numbers don’t take up any room on the number line.”
Mica put her face in her hands and pressed on her eyelids. Eventually, she looked up again. “What other numbers are there?”
“That’s the funny thing.” Saskia’s enthusiasm for her metaphor had risen above that of the question of other time travelers. “Most numbers are transcendental——”
“You’re pulling my leg. Transcendental?”
“That’s what the rest of the real numbers are called. You might know a couple: π, the ratio of any circle’s circumference to its diameter——like how squares are all four times as far around as they are across, but for a circle——and e, the natural base for 100% interest, compounded continuously.”
Mica blinked, unsure where this fantasy would go next.
“Most transcendental numbers, though,” Saskia elaborated, “we don’t have a name for. You’ve likely never encountered them. Of course, π times any algebraic number is transcendental, as is e times any algebraic number, so you know at least twice as many transcendentals as algebraic numbers.”
Mica shook her head, as if to admit that Saskia was speaking another language. “Maybe you should teach someone else to slip in time?”
Saskia’s eyes narrowed.
“I get it,” Mica said. “You’re special, and teaching anyone else would break that. But——”
“It’s not that. It just seems like a bad idea.” And yet, she couldn’t help asking: “Who were you thinking?”
“No one in particular. It only just occurred to me. Maybe someone from Rainbow Earth? You shouldn’t have to take all the physical risks.” She stroked Saskia’s forearm. “I don’t want anything to happen to you again.”
“That’s not exactly a ringing endorsement of the idea to teach others.” It was now Saskia’s turn to scan their neighbors, but no one was paying them any attention. “Besides, I tried taking you and——well we both know how well that went. Hell, you remember your kitchen. We both know I can’t even move an ant. Doesn’t give me a lot of confidence I can impart my powers to others if I can’t even take them on a joy-ride.”
Mica shrugged acknowledgement of Saskia’s point, though a counter-point occurred to her almost immediately. “That might just be me. And the ant.” She raised her eyebrows. “Someone else might even be able to take it another step. Go back faster. Go back farther.”
Saskia’s brow furrowed and she turned back to the window.
Perhaps Saskia’s ability wasn’t uniquely hers. Perhaps she could impart it to others. Whatever the case, she wasn’t excited by the prospect of sharing it with anyone else. At least not yet.
∞
As they reached a cruising altitude, Saskia gazed out over the landscape below. Her trip to Texas had changed her, and her mind wandered in multiple directions at once. She reflected on how lonely of an activity her slipping in time was, and then, mulling her mathematical metaphor, her mind went to Wassily again.
Knowing people across time was an odd phenomenon, that you only appreciated as you got older. It took life lived to experience lives running in parallel suddenly cross paths again. Wassily knew Saskia as another woman. Not just who she was before she learnt to slip in time, but who she was a decade ago. It was the sensation of returning home after years away. Most people first experienced it when they returned home after their first year of college; they’d changed, but done so out of sight. Old friends had a model of them that no longer matched who they were.
What made you you? Was the old you still somewhere inside? Saskia recognized pieces of friends she’d been on the other side of this interaction with. Did we all keep the pieces of our past selves somewhere in the fabric of our person?
And what if that parallel person was another you. How different was her doppelganger who’d left her a note and set out to forge life anew?
Wassily had been an interesting mirror for Saskia, even back in college. He saw who he wanted to see, and he was more interested in her, than she in him. For obvious reasons, his affections were destined to have remained unrequited, were it not for Saskia’s undergraduate experimental curiosity.
Mica caressed Saskia’s hand again, and Saskia gave her a smile. Why was it that she hadn’t come clean to Mica about about her double? Nor that she knew Wassily. Was it a personal character flaw? It was strange to identify a character flaw of her own, one she had never consciously admitted to herself.
∞
The next morning, Saskia had opened her eyes and gazed across at the woman sleeping beside her. It had been late when their flight arrived and it hadn’t taken much for Mica to convince Saskia to stay the night; Pasadena was a lot further for a weary traveler and Mica had been welcoming.
While Mica fixed them both morning chais, Saskia absently pulled a photo album from the bookcase adjacent to the counter that was starting to feel like home away from home. She leafed through a year of Mica’s college life. Camping trips. Some extravagant meals in a communal kitchen. A halloween party.
Turning to a page from spring, Saskia found a photo of Mica, puckered lips presented to the camera. Even as a student hottie, Mica had an air about her of someone who was going to make a difference in the world. Saskia smiled.
Then she noticed the photo on the opposite page. A beautiful young African woman, also with puckered lips. Saskia asked who the other woman was.
Mica grinned. “That’s Wangari. She was gorgeous.” What tickled Mica most about the photo, though, was the way she’d deliberately situated it on the page opposite hers. “We’re perpetually kissing one another, when the album is closed, but whenever someone opens it, that time in life is passed.”
The conceit made Saskia think about branch points. Points where the universe forked, and maybe even folded back into itself. But before she had the opportunity to voice her thoughts, Mica’s front doorbell sounded. Mica’s eyes went wide. “Oh shit. That’s probably Wassily, my mathematician.”
“What?” Saskia felt as flummoxed as Mica looked.
“I forgot I told him he could drop by. He said he had some theory——something about teleporting.” Mica quickly scanned her kitchen for anything inappropriate. Then she looked up at Saskia. “Hey, maybe he’d be a good person to teach time travel to. I already told him about time travel anyway.”
Saskia’s bottom jaw dropped as Mica went to the front door.
“It’s okay,” Mica assured Saskia over her shoulder, “You don’t have to decide now. He’s not staying long.”
