Outdoing Infinity (or: epenis = aleph one)
Is it possible to outdo this? Can you beat infinity? Can your score be higher than toldorknown’s? Yes, it can.
One of the most beautiful pieces of logic (that can be understood by most people) I know of is Cantor’s Diagonal Proof of the uncountability of the real numbers. My first paragraph here was a shameless attempt to get you to read on long enough for me to explain, without scaring you all away with a stack of technical adjectives.
The diagonal proof shows that there is something larger than infinity. It shows that one infinity can be larger than another. It’s quite wonderful. The proof shows this by showing that the set of all integers is, in a meaningful sense, larger than the set of reals. There’s an infinity of integers (-1, 42, 0, 129, -8402975, etc.), but there are even more real numbers. One infinity > another.
The diagonal proof is due to Georg Cantor, a crazy German mathematician who founded modern set theory in the latter part of the 19th century. What Cantor realized was how you can compare sets, even if they’re infinite. If you have a bag of apples and another of oranges, you can easily just count them one by one and compare the counts. But you couldn’t do that if the number of apples and oranges was infinite, because you’d never get to the end. Infinity isn’t a number in any conventional sense; you can’t count to infinity. But you can find out if one infinity of stuff has more stuff in it than another infinity of stuff. Going back to the apples and oranges, you could count them to see if one bag has more fruit than the other. But you could also just take one out of each bag, pairing apple to orange until one of the bags (or both) is empty. That way you could compare the relative sizes of the collections without counting them.
If you could find a method for pairing integers to reals, you could compare them. If we can prove that either there must be at least one integer not paired to any real, or that there must be at least one real not paired to an integer, we know that whichever set has a member “left over” is larger. We now have a new way of comparing stuff: one-to-one correspondences. If a bunch of stuff corresponds one-to-one with another bunch, such that all members of one of the bunches has one and only one partner in the other bunch (no bigamy allowed), and vice versa, we know they’re equally big. And it turns out that you can prove that whatever method you come up with to match integers and reals, there must be at least one real number left over.
But first, something surprising: with this new method for comparing sets, we can prove that there are as many even numbers as there are integers, and as many positive integers as there are integers, even though both the even numbers and the positive integers are what we could appropriately call half of the integers!
The proof consists of finding a method of matching the positive integers or the even ones to the whole set of integers. We proceed from the very reasonable assumption that if you can find a way of matching up pairs of items, one from each of the two sets, so that none is left out and none has more than one partner, then we can conclude that the two sets are equally large. A method of matching the even numbers to the integers is this: take an integer, n, and pair it with 2n. 2n is clearly even, and by this method, we can proceed to match up all the integers with an even partner. No number is left out or counted twice. Beautiful! For pairing integers to positive integers, we’ll interleave negative and positive numbers: 0, 1, -1, 2, -2, 3, -3, etc. We can see that if we keep going, every integer will show up. And we can count this series: for example, 0 is in position 1, 1 is in position 2, -1 is in position 3, 2 is in position 4, and so on. And if we can do this, we can pair every integer on the list with its position in the list, which is guaranteed to be a positive integer. Voila! There are no more integers than there are positive integers, even though there’s zero and negative integers.
We can count the list of integers (0, 1, 2, 3, and so on). This is a property that sets can have: countability. The fact that we can’t count every set is what Cantor’s diagonal argument proves. What do I mean by “can’t count”? Well, there’s always a next number in a countable set, unless you’ve reached the end. We know that the next integer after 1 is 2, and there are no integers between 1 and 2. But what’s the next real number after 1? Is it 1.1? No, there’s always 1.01, 1.07382, and so on. Whatever number we decide is the next number in the sequence, there’s always going to be a number between 1 and that number. (In fact there are as many reals between any two reals as there are reals in total. I’m not going to prove that here; actually, I’ve forgotten how you go about proving that.) The reals are uncountable. Another way of saying that they are uncountable is saying they can’t be put into a one-to-one correspondence with the integers. The set of integers is countably infinite: it’s infinite, but you can count it without ever reaching a point where there’s no well-defined next number. Cantor’s argument is a formal, bullet-proof method of proving that the reals are uncountably infinite: infinite, but you can’t count ‘em.
Recall that we compare the sizes of sets by seeing if they can be put into one-to-one correspondences with one another. Cantor’s diagonal argument runs like this: suppose that we did have a method of mating integers to reals. Suppose we have a table like this:
Real Integer 1 3.14159... -8 9.00000... 45 -0.1980... etc ...
Now comes the great twist: we’re going to construct a real number that is provably different from all the other real numbers in the list; one that, without knowing anything about the method of pairing integers and reals, is definitely not on the list. Do this: for each integer in the list, change one digit of its real partner. Start by changing the first digit. With the next integer, change the second, with the third integer, change the third digit of the real, and so on. 1 becomes 2, 2 becomes 3, 9 becomes 0, and so on. What we get is this:
Real Integer 1 [4].14159... -8 9.[1]0000... 45 -0.1[0]80... etc ...
Take each of the changed digits, and string them together to make a new number. This new number must differ from the first real on the list (its first digit is different), but also from the second (second digit different), the third, and so on. This new number differs from all the numbers on the list in at least one digit. For every n, the number we’ve constructed will differ from the nth real in the list in the nth digit. It’s an orphan! In the Happy Tomorrow Land of Everybody’s Equal, orphans aren’t allowed. The integers and the reals aren’t the same size. There’s always going to be leftover reals, however you try to pair them with integers. One is larger than the other!
(You can object that there could be several decimal expansions for a number, thus violating the requirement that no two members can have the same partner. This can be fixed by making the way we change digits a little more complicated — it’s all been dealt with, it’s just not very interesting to talk or think about.)
Now we finally get to the way to beat toldorknown’s Tumblarity (which is nonsensical anyway, since ∞ is not a number). The cardinality (“size”) of the integers, and all other countably infinite sets, is called aleph null. I’m going to assume this is what the infinity sign is meant to be. The cardinality of the reals, which is greater than the cardinality of the integers (as we’ve seen), is aleph one.
So, yeah. The size of my E-Penis is aleph one. Now you know what that means. And you better be awed.
Interestingly, the word aleph comes from the originally Hebrew letter of that name. I tried pasting that letter into this post, but, being a Hebrew letter — Hebrew is written right to left — it did funky things to the text. I couldn’t figure out how to put a letter to the right of it. So there you have it: I can understand Cantor’s diagonal argument, but when it comes to a measly Hebrew letter, I’m stumped. The Hebrew for alphabet is actually alephbet (אלפבית).
Oh, did I ruin the joke for you? Sometimes, I think, taking a joke seriously is more interesting than not doing so (if not as funny). Did I make a glaring logical error in the above? Surely not! But if you catch me, do let me know.
