We’ve discussed infinity before, but since the subject in inexhaustible, we’re discussing it again.
Infinity comes in sizes, as many readers know. What’s less appreciated is how these sizes relate to questions in epistemology, physics, even theology. We’ll explore some of these facets in future articles. For now, some basics.
The first and smallest infinity is the set of natural numbers. This infinity is not small. All together now: Just how big is it? We are be tempted to say it is incomprehensibly big, and there is some truth to that, but the sticking point is that difficult word comprehensible.
Let’s see if we can get a feel for this tiniest of infinities. Count 1, 2, 3, and keep going. Never stop. Eventually we get to 1010, which is 10 billion. How far is that from the end? Infinitely far away. Take that billion and make it the exponent, i.e. 10billion. How far is that one from the end? Same answer: an infinite distance.
Now that might seem like a lot of zeros after the 10, but it’s a pittance, not even a drop. Exponents are not very handy for counting really large numbers, so let’s work with tetrations. They look just like exponents, except the superscript is on the other side. So 210 = 10^10^10, and 310 = 10^10^10^10, and so on.
We’re getting really big here. Consider B = billion10, which is 10^10^… a billion times. Big number! Bigger than we will ever need for any counting of physical objects. But it’s still infinitely far from the end. Next try BB = B10, which is 10^10^… not just a billion times, but B times. This is so huge it can’t be well thought of. But it’s still tiny compared to the tiniest infinity.
Well, we can keep going, BBB = BB10, BBBB = BBB10, et cetera. Do that B times, then do it BB more times, then BBB more times, and then…you get the idea. You’ll eventually stop, and come to a finite number that is beyond anybody’s ability to grasp (if B isn’t already). But whatever this number is, it is still infinitely far away from the smallest infinity.
What I’m trying to imbue in you is an appreciation how mind-bogglingly big the first infinity is. The number is so large that numbers less than it are all we would ever need if we’re interested in counting (and, yes, many mathematicians call this infinity a number).
After this “simple” counting infinity probably comes what are called, in a playful fit of whimsy, real numbers. I say “probably” because nobody knows for sure if there is another kind of infinity between the counting kind and the so-called continuum, where the reals live. That there is no differently sized infinity between the counting numbers and the reals is called the continuum hypothesis, which most believe is true, but nobody knows how to prove.
Anyway, one way to think about the reals, is to take any two counting numbers, like 1 and 2, and imagine stuffing an infinite number of numbers in between. Count these “stuffing numbers” however you like. Then take any two of these next to one another, still inside 1 and 2, and stuff another infinity of numbers in between them. Like 1.1000000001 and 1.1000000002, and stuff an infinity between them. And keep doing this for any two successive numbers.
You can go on packing numbers into the gaps like this until you come to a point where you have formed a dense flood of numbers, the succession infinitesimally increasing.
The problem, which may be obvious to you, is that if you’re not careful, this infinity doesn’t seem as big as the counting infinity. That’s because it’s impossible, at least for me, to envision what an infinitely dense succession of numbers look like, when I can’t even tell you what BBBBBBBBB10 looks like. Best I can do is to tell you that the number of natural numbers between 1 and BBBBBBBBB10 is infinitely smaller than the number of reals between 1 and 2.
Strangely, counting big natural numbers is hard, yet working with reals is easy. That ease produces in mathematicians a sort of hubris, or rather, forgetfulness. We’re so used to calculating with reals that we forget just how impossibly large the continuum is. The forgetfulness arises when we try and apply real-number equations to things in existence. Are there any actual objects that correspond to the continuum? Depends on what you mean by “actual.”
Skip that question—for now—and think of this. As big as the natural counting infinity is, and as infinitely larger are the reals, there are more infinities larger still. They are comprehensible only in the sense we know they exist, and because we know minimal things about them, such as their ordering. But I don’t think anybody grasps what these numbers are really like. Not when we can’t even say what billion Bs tetrated to a billion BBBB10 is like.
So how many sized infinities are there? And what might this have to do with a proof of God’s existence?
Great question. We’ll do that another time. For those who are adept at math and want to read more, I recommend the paper “Infinite Sets and Infinite Sizes” by the very aptly named Gary Hardegree.