My infinity is bigger than yours
Are there infinities that seem larger but are actually countable?
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The first column I wrote in this space, if memory serves, was about infinity. It was an introduction to the charming but mythical Hotel Infinity, where you can always check in and never leave. Years later, this is infinity revisited, for a slightly deeper examination of what it means.
The first column I wrote in this space, if memory serves, was about infinity. It was an introduction to the charming but mythical Hotel Infinity, where you can always check in and never leave. Years later, this is infinity revisited, for a slightly deeper examination of what it means.
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So,what does infinity mean, anyway? You might think “very large number", but you’ll see it’s more than that. You might think “numbers without end", but then ask, exactly what does “without end" mean? What is there around me that has no end, that doesn’t ever finish?
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Questions, questions!
Still, a good way to start thinking about infinity, at least with numbers, is that there is no end to them. Start counting 1, 2, 3 ... and if you don’t die or are otherwise halted, you will never stop. Or look at it like this: No matter how large a number you name, there is a larger one, and the simplest way to get a larger one is to add 1.
Think a billion is the largest number there is? I give you a billion and 1. “Oh yeah?" you shoot back, “what about 987,168,940,050,370,281?" I give you 987,168,940,050,370,282.
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You get the idea. That little step of adding 1 to get a higher number, of always being able to do so, is really infinity beckoning to you. And once you respond, there’s all kinds of intriguing things to explore. Some accomplished mathematicians have trod this path.
Start here. If you count 1, 2, 3, 4 ... and never stop, and that’s your first glimpse of infinity—well, you could also count 2, 4, 6, 8 ... —just the even numbers, and you’d never stop with those either. Does that seem strange? After all, in counting only the evens, you’re dropping the odds. You’re leaving out fully half the whole numbers we have. Yet that half is infinite.
You could leave out many more numbers too. Consider:
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* 1, 1001, 2001, 3001 ... skips over a thousand integers at a time.
* 1,000,000, 2,000,000, 3,000,000 ... omits a million integers at a time.
Yet both of those sequences, sparse as they are, are infinite as well.
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And then consider this:
* 1, 10, 100, 1000, 10,000 ...
Between its 7th and 8th terms, this sequence skips past 9 million integers. With each term, it gets ten times more sparse. Yet, this sequence is infinite too. Think of what’s going on here. There’s a qualitative difference between the last sequence and the ones I mentioned earlier. Those had a fixed gap between each term, but this one has gaps that increase ten-fold with each term. Yet all are infinite.
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In fact, you will agree with this assertion: they are all infinite in the same way that 1, 2, 3, 4 ... is infinite. In each case, if you happen to stop somewhere along the line, you can easily say what the next term is and pick up from there.
Beyond intuition, there’s actually a serious mathematical concept buried here. Remember we started with the numbers whose sequence is most natural to us: 1, 2, 3, 4 ... so much so that we call these the “natural numbers". We use these as serial numbers all the time—for example, my first “D" in college, my second “D", my third “D", and so on. (I had a lot of Ds)
Now, take each of the other lists I mentioned above. You can label each of the terms with a serial number. Considering the even numbers, the first is 2, #2 is 4, #3 is 6, etc. Considering the millions, #1 is 1,000,000, #2 is 2,000,000, #3 is 3,000,000, and on from there.
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We say that there is a “one-to-one correspondence" between these pairs, because for each term in 1, 2, 3, 4 ... there is a corresponding term in the other. It’s even clearer if I write one of these sequences—take the powers of 10—like this:
#1: 1
#2: 10
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#3: 100
#4: 1,000
#5: 10,000 and so on.
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When you have a one-to-one correspondence like this—when every term can be paired with a corresponding term in the sequence 1, 2, 3, 4 ... —suddenly, we have a way to come to grips with the infinite. It was the brilliant German mathematician Georg Cantor (1845-1918) who first cut through the mystery infinity was always wrapped in and gave us a measure of clarity. Because of him, the infinite is now just another facet of numbers and mathematics.
Cantor called a sequence like those above a “countable infinity". Meaning, you won’t ever reach its end, but you can actually count its terms. In Cantor’s view, all countable infinities are the same “size"—because they have that one-to-one correspondence with the natural numbers.
Though as soon as you mention size, you wonder—are there smaller ones? Bigger ones? Cantor puzzled over that, and concluded that countable infinities were the smallest possible infinity. He used the first letter of the Hebrew alphabet, Aleph, or À to label them: Aleph-0, or Aleph-nought. Then he asked two questions: one, are there infinities that seem larger but are actually countable, meaning they too are Aleph-nought infinities? Two, what does a “larger" infinity look like?
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His answer to the first gives you a glimpse into the mind of this man. Consider the fractions—meaning numbers that can be expressed by dividing one integer by another (like 1/5, or 355/113). Intuitively, it’s easy to think there must be many more fractions than natural numbers. But here’s Cantor, telling us to list the fractions this way:
1/1
2/1, 1/2
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3/1, 2/2, 1/3
4/1, 3/2, 2/3, 1/4
5/1, 4/2, 3/3, 2/4, 1/5
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6/1, 5/2, 4/3, 3/4, 2/5, 1/6 ...
This sets up a clear and unmistakable order you can use to count the fractions. A little thought will tell you that every fraction you can dream up will figure somewhere in this table. Where’s 355/113, you ask? Add the numerator and denominator to get 468, then go to the 467th row and step right 112 steps—there it is. (As it happens, the 108,924th entry in the table).
Indeed, the fractions are countable.
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But if those are the fractions, Cantor showed that the decimals are uncountably infinite, meaning they cannot be put in one-to-one correspondence with 1, 2, 3, 4 ...
He proved this with another flash of brilliance. Assume they are countable, he said. Then we should be able to list them somehow. For example, this might be your list:
#1: 0.14016613750...
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#2: 0.34615107137...
#3: 0.52147862734...
#4: 0.47604876905...
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$5: 0.26427793981...
Now create a new decimal by taking the first digit from #1 and adding 1, the second from #2 and adding 1, the third from #3 and adding 1, etc. We get:
0.25213...
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Again, a little thought will tell you that this number cannot exist in your list. Precisely because we built it the way we did, it will differ from every number in the list by at least one digit—that one we added 1 to. Therefore, our purported list of fractions is incomplete. Therefore, the fractions are uncountably infinite—in fact, infinitely larger than the countably infinite natural numbers. Cantor called this kind of infinity Aleph-one.
There’s a lot more here to explore. I’ll leave you with something else Cantor discovered: that once you start chasing infinities—his beloved Alephs —they don’t stop either.
There’s an infinity of infinities. Isn’t that just perfect?
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Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun
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