by Chris Bodenner

I have an explanation for Zeno’s paradox that makes it easy for your readers to understand the infinite sum. Zeno's paradox arises from *stopping* an infinite process and declaring on that basis that the process can never finish, i.e., you never cover 100% of the room’s length because you stop at some point and say there is some distance that has yet to be covered, and conclude that it therefore *never will be*. An easier way to picture this infinite process, at least for me, is to arrive at the result backward.

We know that the room has a length. Let’s call it L. We also know that we can divide the room's length into two equal halves, call them A and B.  (A + B = L by definition.)  We can repeat this process on the second half, that is, we can divide B into two equal halves. These two halves must add up to B. Now imagine carrying out this process infinitely. Is it not obvious that the grand total of all these lengths must add up to the room length L? The only difference between my "backward" approach and Zeno’s approach is that you can stop mine at any time and see that the total must equal L. However, the two approaches are, mathematically, exactly the same sum. Therefore, one can conclude that Zeno's approach, if carried out to infinity, would allow one to cross the room and touch the farthest wall.

Another reader pulls from the previous post:

Another needs a translator:

Hilbert's example only uses countable infinity, or infinity where there is a one-to-one correspondence with whole numbers, or positive Integers.  This is only the first order of magnitude (cardinality) of Infinite Sets, dubbed Aleph-Null (ran outta Greek letters and went to Hebrew).  Next order of magnitude would be Aleph-one – the set of Real Numbers for example (provable depending on the Axiom of Choice and other stuff I don't quite remember), which includes the irrational numbers like pi, e, SQRT (2), etc.  Go here, here and here for some fun (well, for some of us, anyway) reads. A bit easier to figger out: Zeno's paradox, explainable with calculus (convergent infinite series).

You rang?

This is actually my favorite piece of math to explain to laymen. It's mindblowing and quite understandable with a little effort. (Although I've included a couple "bonus" facts in this email that I didn't really prove and don't expect anybody to fully understand.)

First fun fact: there is a "one-to-one correspondence" between the natural numbers (1,2,3, …) and all the integers, meaning in a certain sense there are no more of one than there are of the other. I'll show you that correspondence right now:

1  2  3  4  5 ….
0 -1  1 -2  2 ….

Any time a set of numbers can be placed in one-to-one correspondence with the natural numbers, we call it "countably infinite." That's because one-to-one correspondence is exactly how we normally count objects: "This is apple number 1, this is apple number 2. Oh look, we're out of apples. There must be only two apples." As your previous reader pointed out, the symbol for "countable infinity" is Aleph-null. (That's Hebrew letter "Aleph" with a subscript of 0).

Okay, that was an easy one. Here's something a little more mindblowing. The set of all rational numbers (fractions of integers) is still countable! It's no "larger" than the set of natural numbers, in this sense … in spite of the fact that there are infinitely many fractions between any two natural numbers! The proof is easy, but to save space I'll just throw in a link. Its diagram tells the whole story:

Just follow the arrows and number the fractions in order as you go. (Okay, this diagram only covers the positive rational numbers, but that could easily be rectified.)

So by now you'd probably be surprised if I told you there IS something so big it isn't countable! Well, there is: for example, the set of real numbers (i.e. anything that can be represented as a decimal, infinite or non-infinite).

I'm going to prove that the real numbers between 0 and 1, alone, are so numerous that you can't put them in one-to-one correspondence with the integers. The way I'm going to do this is a little sneaky. I say, "You show me a list of real numbers in one-to-one correspondence with natural numbers, and I'll show you a real number that isn't on your list. No list you give me can possibly be complete! It's impossible!" So let's get to it. Here's an example list:

1: 0.14260000…
2: 0.36378423…
3: 0.55922342…
4: 0.12345678….
etc.

To find my counterexample, I'm going to take the first digit of the first number, the second digit of the second number, etc.:

0.1694….

and add 2 to each one (turning 9 into 1, etc):

0.3816….

Now, this infinite decimal I've constructed isn't "decimal number 1": it has the wrong first digit! And it's not the second number in the list either: it has the wrong second digit! In fact, it isn't ANY of the numbers on your crappy list! And I can do this no matter what list you give me. There are just too many real numbers for you to try this game with me.

Evidently the collection of real numbers is larger than the sets we've been dealing with before. We'd like to give a name to this new type of infinity. We could call it "Aleph-1" but that seems to imply that there are no sizes in between this and Aleph-0. So maybe we should first decide whether that is true or not? Rather fascinatingly, it all depends: you can actually construct entire logically-consistent systems of math where it either is or is not true. Of course this means that even if there's a set smaller than the real numbers but larger than the natural numbers, it is impossible for me to explain to you which numbers are in that set and which aren't! That would constitute a proof of its existence, and yet its existence can't be proven without breaking those mathematical systems in which it doesn't exist!

Anyway to close out this explanation, I'll recall that another reader pointed out that there are infinitely many different sizes of infinity. But wait, that's not good enough for us anymore, is it? Are there countably many types of infinity, or more? Could we call them all "Aleph 0," "Aleph 1," "Aleph 2" and so forth, or would that only scratch the surface? Turns out the answer is "It's not countable. In fact, it's larger than any "size" of infinity that we can possibly construct."

If your brain isn't fried at this point, another:

As a mathematician, I must point out the error in the previous post about infinities which you said "needed a translator". The writer points out that there are indeed different "sizes of" infinities. (I present a "humanized version" of the argument below). But he's wrong when he says that the next infinity after Aleph-Naught is ("assuming some other things") provably Aleph-One.

Math is a study which begins with axioms and then derives logical conclusions (theorems) based on those axioms. The question is, "What are the most fundamental axioms?" Euclid put forward his five (the famous Parallel Postulate among them), but these aren't used any more. Peano put together his, but these are also not used any more as a basis of mathematics. Instead, the standard set of axioms were laid down by Zermelo and Frankel and concern the question of "What is a set?", the most basic of all mathematical objects. To illustrate why you have to think carefully about this question, ponder Russel's Paradox (invented to dispatch Frege's attempt at axioms), which begins: consider the set S of all sets which don't contain themselves. Then S belongs to S if and only if S does not belong to S. That is, "if it does, then it doesn't, and if it doesn't, then it does." This, being pure logic and not a debate among politicians, is impossible.

As you can see, this level of math gets rather mindbending very fast, and almost all mathematicians eschew the study of such things. Most of our definitions are at a level of abstraction much further up than these "basic considerations" (partly because we don't understand them). Indeed, it is here that your reader made a mistake. The statement that the "next infinityafter Aleph-Naught is Aleph-One" goes by the name of the Continuum Hypothesis. Godel and Cohen have shown that this statement is independent of ZFC, the standard set axioms. That is, assuming these axioms don't themselves contain a contradiction, it can be neither proven nor disproven from these axioms alone.

As this is where my head starts to hurt, I'll leave you with humanized version of Cantor's diagonalization argument:

Suppose there were two infinite hotels next to each other, the Hilbert Hotel and the Cantor Hotel. At the Hilbert Hotel, the rooms are labeled by the natural numbers: 1, 2, 3, 4, …. At the Cantor Hotel, the rooms are labeled by the real numbers, which include the natural numbers, the rational numbers (fractions), and all the irrational numbers such as the square root of 2, pi, e, the logarithm of almost anything, etc. Further suppose that Hilbert's hit on hard times so his hotel is empty. But Cantor's struck it rich and every single room is currently occupied. That is until a massive fire burns down the entirety of Cantor's Hotel. Everyone survives, and so they ask Hilbert if he has room for them all. Does he?

Well, suppose he did. Then there is a correspondence between the tenant's room number at the Hilbert Hotel and their room number at Cantor's Hotel. It's hard to write down real numbers, but we know that every real number has a decimal expansion (pi = 3.1415926…, e = 2.718281828459045…). So for our labeling purposes, let's forget about the stuff to the left of the decimal (the "whole part" or "floor") and just worry about the stuff to the right of the decimal (the misnamed "fractional part"). So the correspondence between room numbers looks something like:

1 —> ….2547348….
2 —> ….3453839….
3 —> ….2837384….
4 —> ….3824935….

Now to show that Hilbert *cannot* hold all of Cantor's guests, we just have to find one person who's out in the stables. So let's try to find someone (who we will identify by their room number at Cantor's Hotel). We construct a number as follows: it will have 0 whole part. The first decimal digit of its fractional part will be any digit other than the first decimal digit of the fractional part of the first person on our list. That is, person ….2547348…. So we choose any number other than 2, say 3. The same goes for the second digit of our new number: we choose any digit other than the second digit of ….3453839…., say 3.

We continue in this way for every room at Hilbert's Hotel and we get the number 0.3343…. Now whoever was in this room at Cantor's Hotel is not in Room #1 at Hilbert's Hotel (since 0.2547 does not equal 0.3343), or at Room #2 (since they differ in their second digits by construction) or Room #3 (the third digit differs) or #4 (the fourth digit), etc. That is, this supposed correspondence between rooms at Hilbert's and Cantor's Hotels is missing 0.3343….

Conclusion: Somehow Cantor's Hotel is bigger than Hilbert's Hotel.

Time for a mental health break:

A smaller bite to chew:

And we can take Cantor and Hilbert and take it one step further!  As you're currently reading Gödel, Escher, Bach, you'll appreciate this. As a prior reader noted, there are an infinity of infinities, denoted as aleph-null, aleph-one, etc.  That lead to the next obvious question: how do we know that there are no infinities between aleph-null and aleph-one?

Mathemeticians couldn't prove it one way or the other, but they were able to at least get halfway there, and proved the following: if we assume there are no infinities between aleph-null and aleph-one, it wouldn't violate any axioms. But then somebody proved the opposite: if we assume there *are* infinities, between aleph-null and aleph-one, it wouldn't violate any axioms.

So, then came Godel who looked at the statement: "the statement that there are infinities between aleph-null and aleph-one is unprovable" and wondered: what else is unprovable? Eventually came Godel's first Incompleteness Theory, which, simplified, says that *every* logical and consistent system will have assertions which can not be proved. This is a great example of how math moves forward – discovering things, asking questions about it, generalizing on it, and repeating the process.

A math major writes:

Sometimes people who are really good at math tend to be crazy.  Gödel himself was an example.  I was a good math student, at the top of my college classes, but not someone who was destined to go out and make new discoveries.  I am a little crazy sometimes, though.  I think it has to do with pattern recognition.  I imagine a system in people's brains that makes connections between patterns in one context and patterns in another.  People who are really good at math tend to have the volume knob on that system turned up a little higher than other people do.  This is great for spotting things in the theory, but it also tends to cause them to imagine connections in the world at large that aren't always there.

Another:

The one that gets me is hearing Dr. Neil deGrasse Tyson talk about infinite space. I first heard him on this topic on NPR's Science Friday, but can't quickly get my hands on that link. Here is video of him discussing it on ABC (this topic comes up at 3:45, earlier he mentions the different kinds of infinity your reader wrote about). My head asploded when he said that if there are a certain number of ways that particles could be arranged, and space is infinite, then what we think of as "parallel worlds" (say, another Earth but one in which I'm not sending you an email but went for more coffee instead) are physically out there. Travel far enough and you will run into the same patterns of particles with slight variations over and over. For infinity.

Yet another:

I am sure you are aware of this, but D.F. Wallace has a great book on this subject, specifically George Cantor's proof, A Compact History of Infinity. I have read it several times, and while the math is a bit out of my range, the writing and explanations of the complicated mathematical concepts are great, as is typical of DFW's nonfiction.

Almost done:

You might want to mention that the late and so-much-missed David Foster Wallace wrote a very enjoyable little book on the history of the mathematics of infinity.  Everything and More is eminently readable, easily comprehensible to someone with only a smattering of calculus, and contains the usual nuggets of Wallace's profundity and humor.

A reader sent the above video:

While Cantor's take on infinity is interesting, I remember the headache it gave me when I took my first course in set theory. Another way your readers may enjoy contemplating the concept of infinity is to watch this video on imagining 10 dimensions. It's meant to help understand some of the concepts of multi-dimensional space as used in thing like string theory, but touches on the concept on infinity and how there can be different infinities.

The gist of it is to realize that for any given starting point of the universe (say a particular set of laws, like how strong gravity is, or what the speed of light is), there are an infinite number of ways that universe could play out. The next step is to realize that there are an infinite number of different starting points, each with an infinite number of outcomes. That is, there are an infinite amount of infinities.

I think the video does a nice job of building this up from the very simple starting point of just imagining a single dot. Some people may get a little lost as it gets into the higher dimensions, but after a few viewings, most people should be able to understand it pretty well. Plus, it shows that once you imagine all the possible branches of all the possible time lines of all the possible universes (ie, an infinite amount of infinities), you eventually do hit an end to this train of logic. That is, the brain finally is able to get some rest upon realizing that there is, in some sense, an end to the endless.