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In my previous installment, I attempted to establish that mathematics is a sequential art. I think it’s already time to get meta: sequential art about sequences.

All of the works you think of when referring to “sequential art” deal with finite sequences. At some point, the panels stop and the comic ends. Let’s talk a little about the loss of intuition, vague sense of the world spinning, and loss of appetite that comes with infinite sequences.

One of the first popular notions of the problems with infinity was also the introduction of my favorite name for an argument, reductio ad absurdum, i.e. proof by contradiction. Zeno’s paradoxes, namely the “racetrack paradox”, purports (in Aristotle’s words) that “motion is an illusion” and that, therefore, all Reality is indivisible.¹

The racetrack paradox goes as follows:

If you want to walk 100 steps, you need to go 50 first, but to go 50 you need to go 25 first, but then you need to go 12.5 first…

A runner is about to start a race. Let's assume the finish line is 1 km away. Before he can get to the finish line, he has to reach the halfway point. Of course, he has to get halfway to that point first -- 1/4 of the way... and so on. Basically, since Zeno wants our runner to travel the distance of the infinite sum

which is the sum of the infinite sequence

The problem is, Zeno wants to sum from the "last" term first, and since there are an infinite number of terms, he can't do that. Therefore, motion is an illusion, and All is One -- there are no "different" objects.²

Part of the confusion that occurs when you think about infinite sequences is in grasping the notion of a limit. The immediate intuition is that you have to somehow travel, physically, through an infinite number of points before you reach your destination (the end of the story, say), and, like Zeno's runner, since you "can't do that", then... it just doesn't make sense, okay?! Not exactly. The problem is one of distinguishing between "goes to" in a time sense and "goes to" in a logical sense. A convergent infinite sequence or sum just is; it doesn't go anywhere.³ The terms just have to be "close together" enough further out in the sequence. Logical narrative versus time narrative. Mathematicians know that, abstractly, a sequence that converges already is converged, and the answer exists.⁴ You don't have to collect your hobbit friends and follow an infinite trail of bread crumbs to find the One True Limit of the Convergent Sequence. It's just there. Click your heels together.

... or something like that.

Usually we start a little slower than that. One of the first math questions most people have is, "what's the biggest number?" They're confused and frustrated when told that "there isn't one," the numbers just "keep going" (even though they're not going anywhere, they're just there. Abstractly.). Counting is the first explicit sequence we learn: 1, 2, 3, 4, 5, .... and as we know, it doesn't end. We call these numbers the counting numbers (or natural numbers) for that reason.

There are as many even natural numbers as natural numbers (which is of a certain "size" of infinity called countably infinite):

and the sequence keeps going... even though it just "makes sense" that there are more whole numbers than just even whole numbers? (Like twice as many?!) On top of that, there are the same amount of fractions as there are natural numbers. It all has to do with a thing called one-to-one correspondence. Make a numbered list (i.e. a sequence) of your things. If they all fit on a list, then your stuff is countable. You "write down" an infinite sequence by some argument.

Try Hilbert's Grand Hotel. Hilbert's idea: the hotel has an infinite number of rooms, and it's always full. If one new guest shows up, just tell everyone to move down one room, and the new guest can have Room 1. If a countably infinite number of new guests show up, just shift all the current guests to double their current room number, and stick all the new ones into the now-vacant odd-numbered rooms. Who cares how "long" it takes? This is a Gedankenexperiment, logical narrative versus time narrative! It doesn't even "happen" instantly - it just is.

"Hilbert's Hotel", Carol Lay

Probably the most difficult one for an undergrad mathematician (hell, anyone) to handle is the fact that the set of real numbers, which has been constructed in a few different rather boring ways, all dealing with infinite sequences, is a bigger infinity of size than the whole numbers. (Yes, Virginia, there are different sizes of infinity.) This one is most popularly proven by Cantor's diagonalization argument.

The diagonalization argument basically says: Say you've put the real numbers on a list. Each is an infinitely-long string of digits. Now, take the nth digit from number n on the list and add one to it (change 9s to 0s if you need).

"Doing" this infinitely, we've built a number that cannot be on the list - it disagrees with the first number in the first digit, the second number with the second digit, and so on, infinitely. Therefore, there are more numbers on the list than "countably infinite". Reductio ad absurdum.

If this doesn't seem to make sense, there are always arguments for the existence of God (the same being that Cantor thought chose him to provide proof of different infinities through sequences). I'll leave those to you. Reductio ad absurdum.

"No one shall expel us from the Paradise that Cantor has created."
- David Hilbert

[1] It's unfortunate how many Greek works are known only secondhand - Pythagoras, Zeno of Elea, Socrates... the list goes on. It is, of course, a finite list.

[2] Nobody, even then, said he was right about this, but it did cause some philosophical problems for a couple millenia.

[3] Of course, there's the omnipresent 0.999... = 1 argument. No way am I getting into that. It just is, okay?

[4] This is why computer scientists hate us. And physicists, and engineers, and the general public..........

[5] I would not want to go grocery shopping for Cantor.