Videos: A taste of infinity

A taste of infinity

16 October 2009

(Note: The video contains various diagrams and notations that are helpful in understanding the material.)

ZJ: Infinity. Eternity. Endless. Forever.

When we use these terms, do we really understand what we're talking about? Certainly we know what they mean: that which has no boundaries, that which is without limit, that which does not stop at any point. But do we actually have a practical sense of what infinity really is? Do we have a meaningful grasp of infinity itself, in the same way that we understand what an hour means, or a century, a kilometer, or a light year?

No. True infinity is simply incomprehensible to us, because everything in our reality is limited—including reality. The size of the observable universe and the duration of time itself, though increasing, are both finite quantities. Nothing that exists is infinite, so infinity just isn't something we can relate to in any useful fashion. Whatever our concept of infinity is, it will inevitably fall short.

Perhaps that's why some people are so casual about infinity, using it for utterly trivial purposes. They ascribe attributes like infinite existence and infinite wisdom to beings that are entirely imaginary. They claim we will all experience an eternity of absolute bliss, or unfathomable suffering, depending on something so insignificant as whether or not we believe in these beings.

And yet, they really have no idea of what eternity entails, because everything in their experience, however pleasurable or painful, is of a finite duration. It seems that when we don't properly appreciate the meaning of infinity, we can easily find ourselves throwing it around for the most frivolous reasons, as if it were meaningless.

If we want to have a slightly more realistic sense of infinity, perhaps we should try thinking about a number that is not infinite, but still very large. Let's contemplate Graham's number.

Graham's number was first conceived of by Ronald Graham, who used it in a mathematical proof in 1977. It is so large that conventional notation cannot describe it. To represent Graham's number, we need new operators. These operators start with exponents. Exponentiation is repeated multiplication. A base is raised to an exponent, meaning the base is repeatedly multiplied by itself, and the exponent determines how many times this is repeated. Let's use a single up-arrow to indicate exponentiation. For example, 3 to the power of 3 (3 ↑ 3) would be 27.

Now suppose we create a new operator to indicate repeated exponentiation, in the same way that exponents are repeated multiplication. Let's use two arrows for this, as in, 3 double-arrow 3 (3 ↑↑ 3). This is 3 to the power of itself, 3 times (3 ↑ 3 ↑ 3). Remember that exponents are calculated from right to left (3 ↑ (3 ↑ 3)), so we have 3 to the power of 27 (3 ↑ 27), which is about 7 trillion (7,625,597,484,987). This is not a large number at all.

But let's create another operator, ↑↑↑ (triple arrows), for repeated use of ↑↑ (double arrows). For example, 3 ↑↑↑ 3 would be 3 double-arrowed by itself 3 times (3 ↑↑ 3 ↑↑ 3). Just as with exponents, each of these arrow operators are calculated from right-to-left (3 ↑↑ (3 ↑↑ 3)). As we've seen, 3 ↑↑ 3 is 7,625,597,484,987, so 3 ↑↑↑ 3 equals 3 ↑↑ 7,625,597,484,987. This means 3 to the power of itself 7,625,597,484,987 times. That is, a stack of threes 7,625,597,484,987 high.

Just think about what it would be like to work our way down from the top. Starting with 3 ↑ 3, we get 27. Then we have 3 ↑ 27, which is 7,625,597,484,987. Then we have 3 ↑ 7,625,597,484,987. And then we have 3, raised to the power of 3 to the power of 7,625,597,484,987 (3 ↑ (3 ↑ 7,625,597,484,987)). And those are just the first 5 threes, in a stack of 7,625,597,484,987 exponents. Think about how quickly these numbers became so immense, in only the first five terms, and imagine repeating this 7,625,597,484,987 times. That is 3 ↑↑↑ 3.

At this point, any comparisons involving the number of grains of sand on every beach on earth, or even the number of atoms in the universe, have no meaning whatsoever. They are nothing next to the number produced by 3 ↑↑↑ 3. But even though the number itself is beyond our comprehension, we can still understand how it was created. The concept of a series of 7,625,597,484,987 threes can still fit in our heads. But we haven't arrived at Graham's number yet.

Now let's use ↑↑↑↑ (quadruple arrows) for repeated use of the ↑↑↑ operator. So, 3 ↑↑↑↑ 3 would be 3 ↑↑↑ 3 ↑↑↑ 3, or 3 ↑↑↑ the number created by a stack of 7,625,597,484,987 threes. This comes out to 3 ↑↑ by itself, 3 ↑↑↑ 3 times, which is far past anything that could be expressed as a stack of exponents. We cannot even imagine how large this number is.

And consider how few steps were needed to reach this point. 3 ↑ 3 is 27. 3 ↑↑ 3 is 7,625,597,484,987. 3 ↑↑↑ 3 is 3 to the power of itself, 7,625,597,484,987 times. And 3 ↑↑↑↑ 3 is just unbelievably larger than 3 ↑↑↑ 3. We've come this far, with only four arrows. And we can keep going, defining operators with more arrows, and each of them will indicate repeated use of the operator with one fewer arrow.

Now that we've seen how rapidly these numbers increase from adding only one more arrow every time, let's move on to Graham's number itself. We'll refer to 3 ↑↑↑↑ 3 as g₁. It will be the first term in a sequence. g₂ is defined as 3, followed by g₁ arrows, followed by 3. The unthinkably huge number g₁ is how many arrows are in g₂.

But it gets even bigger than that. g₃ contains g₂ arrows, and this is repeated until we reach g₆₄. And g₆₄ is Graham's number (G).

The immensity of this number is simply inaccessible to me. I cannot wrap my mind around it. Even the first few steps of Graham's number put infinity in a completely different light for me. Initially, I imagined infinity to be very large. But I didn't know it was that large. Even 3 ↑↑↑ 3 was bigger than my sense of infinity. And once I got to g₂, just thinking of how monstrous that number must be, with so many arrows, was just overwhelming.

Graham's number is so far beyond anything in our reality, even the infinitesimal portions of it that we can understand are horrifying, like catching a glimpse of something that simply cannot be, something incompatible with our world. Its enormity overtakes us, reducing everything that exists, and anything we could imagine, to nothing at all.

So, if you ever feel like talking about infinity, try thinking about Graham's number instead. Try to comprehend as much of that number as you can. And then consider whether infinity is what you really mean.

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