Tuesday, September 11, 2012

Aleph and Tranfinite Arithmetic

Aleph is the first Hebrew letter.  By way of the Phoenician aleph, it is associated with the Greek alpha, and from there to the Latin/English A. It corresponds closely to the Arabic letter alif.  It is said to mean "ox".  It's associated with the tarot card The Fool, and traces the path from Keter to Chockmah.

It is difficult to describe the sound it makes.  Open your mouth, begin to say the word "all", but stop before you actually say anything.  Or, say "uh-oh" (as if something bad happened).  Aleph is the sound between the two syllables.

As expected, it is associated with the number 1.  Aleph begins the word El, which means God, as well as the words Ehyeh (I am), Emet (truth), and Achad (One/Unity). Aleph is very closely associated with the element of air, most especially with breath.

Examine the way the aleph to the left is drawn.  See that there are two mirror-image pieces(1) in the upper right and lower left, separated by a line?  This represents how G-d is both immanent (manifest in creation) and transcendent (beyond creation).  It can also be seen as a telling of how the created world is a reflection of the creator and vice versa (as above, so below).

There is a story told, that when G-d created the universe, he assembled all the letters and asked who should be used to begin Torah.  The aleph demurred modestly, and so was given the honor of beginning the Ten Commandments.  (I'll tell more of the story later...it's really a story about beth.)

I associate the aleph with the opening chapter of the Dao de Ching: "The way you think you know is not the Way.   The name which can be said is not The Name.  The Aleph is the beginning of Creation, but by naming are the myriad things created."

For me, aleph represents the notion of infinite creation seen as a single unified thing.  This links it intimately with the Shema:
"Shema Yisroel, Adonai Eloheinu, Adonai Echad"
"Hear, Israel, G-d is Our God, G-d is One."
"Listen, Israel:  G-d is both our patron god (2), and also the Ineffable One."

When I meditate on aleph, I do so mathematically.  In mathematics, the aleph stands for the set of all the transfinite cardinalities. That is, it is the set which contains all the different sizes of infinity.

Each different size of infinity has an aleph in its name.  For example, the size of the counting numbers is aleph-0.  (said "aleph-naught")  Anything which is this size is said the be "countable".  Things of this size can be listed, although the list goes on forever.

To understand this size of infinity, imagine a hotel with an infinite number of rooms, but which is entirely occupied.  If a new guest arrives, can the clerk find her a room?  Yes, but it is tricky.  The clerk makes an announcement, that each guest must pack up and move into the room to their right.  For example, the guest in room 1, must move to room 2, the guest in room 2362 must move to room 2363, and so on.  In this way, room one will be available.

Another size of infinity is the number of points on a line.  This is much larger than the number of counting numbers (aleph-0). 

UPDATE: There's a new, better explanation of this here: http://traifbanquet.blogspot.com/2015/03/on-infinite.html

There are many ways to prove this, but the easiest is by contradiction.   First, we will start with a line segment 1 unit long.  It should be clear that the number of points on this segment must be less than or equal to the number of points on a whole line, which extends infinitely in each direction. (In fact, they contain the same number of points.)  With each point on our unit segment, we can associate a real number between 0 and 1.  For example, the point 1/2 of the way along the segment we denote by 0.50000000000000...  The number a third of the way along is point 0.33333333...

Now, if there are the same number of points on our segment as there are counting numbers, we should be able to match them up one-to-one.  That is, we should be able to write a list of the points' numbers.  I will show that it is impossible to have such a list by showing that, no matter how the list is ordered, there will always be at least one number between 0 and 1 which isn't on the list.  Let's say our list looks like this:

1: 0.03426...
2: 0.31964...
3: 0.48612...
4: 0.12694...
5: 0.00971...

Now, I will construct my number by first highlighting the diagonal along my list.  That is, I pick out the first digit of the first number, the second digit of the second number, etc.  In our example, this number begins: 0.01691...

Now, I will construct a number number by adding one to each digit.  In order to keep things circular, let 9+1=0.  In our example, this new number begins: 0.12702...

I claim that this number does not appear ANYWHERE on the original list.  This is how I know:  In order for two numbers to be the same they have to match in every decimal place.  That is to say, if two numbers differ in even a single place, they are not the same number.  The last number we constructed differs in at least one place from every other number on the list.  Its first digit is different from the first digit of the 1st number; its second digit is different from the second digit of the second number; its thousandth digit is different from the thousandth digit of the thousandth number; and so forth.  So, we know that there is a number which isn't on our list.  Since this would have worked for any list, we conclude that there are just too many points along a line to list (even an infinite list), and so the number of points on a line must be BIGGER than the number of counting numbers.

While I will not prove it here, in fact the number of points on a line is exactly equal to 2 to the aleph-0 power.  (Where, recall, aleph-0 is the size of the counting numbers.)


(1) These pieces are actually the letter yod.  I'll talk about that more when I get to yod.
(2) Like Athena for the Athenians.


  1. Have you read the short story "The Aleph" by Jorge Luis Borges? It talks about the infinity and its wonders and the darkness in our hearts.

  2. Checking out of the hotel must be mightily complicated. What if the person in room 2363 wants to leave?