There Is More than One Infinity

There Is More than One Infinity February 3, 2013

I have unfortunately seen some people argue that, “Since the Christian God is infinite, our Pagan Gods must be finite.” Um, no. I don’t want to be snarky, but that is just REALLY bad theology. One cause of this problem is the assumption that there is only one infinity and that Old Jehovah Nobodaddy has an exclusive claim on it. Again, no. There is more than one infinity. In fact, there are an infinity of infinities, of an infinity of different types.

I know it doesn’t occur to many people, even if they understand what theology is all about, that theology can be done mathematically. It can be. Just watch.

In daily life, it is hard to remember that “infinite” means not just “very big,” but “unending.” There is, happily, one unending set we are quite familiar with, that of the natural numbers: 1, 2, 3, 4, 5, etc. About a century ago, a German mathematician, Georg Kantor, thought maybe he could describe infinity in a little more detail than just “unending.” He began by asking “How many natural numbers are there?” He proposed the concept of “The number of all numbers,” which he called a “transfinite” number and symbolized as א0 [that’s supposed to be a large Aleph followed by a subscript zero, which it was in my text, but this system is not sophisticated enough to handle mathematical notation correctly, and I’m not going to mess around with it any more; the notation is pronounced “aleph null”]. Next, Kantor pointed out that “to count” means to place the items in the set we are counting into one-to-one correspondence with the set of natural numbers: 1, 2, 3, etc. The number we reach for the last item tells us how many items are in the set.

Next, let’s ask an apparently dumb question: How many even numbers are there relative to all the natural numbers? Isn’t it common sense to think there only half as many? But if we count the even numbers by placing them into one-to-one correspondence with all the natural numbers, which is our ordinary “counting rule,” what we see is:

2          4          6          8          10 . . .

1          2          3          4          5 . . .

Obviously there are just as many even numbers as there are natural numbers; so the number of all even numbers is also א0. Equivalently, aleph null divided by 2 equals aleph null.. Hence any defined subset of the natural numbers, such as all the millions, will also equal א0. You can already see that this line of thought is going to look steadily more paradoxical.

Furthermore, the number of all rational numbers (fractions) also equals א0. To see this, set up a coordinate grid, with numbers 1, 2, 3, 4, etc., along the x-axis and down the y-axis. In the grid, the item at each intersection will be a fraction, with the x value on top and the y value on the bottom, like so:

0          1          2          3          4          5 . . .

1          1/1       2/1       3/1       4/1 . . .

2          ½         2/2       3/2       4/2 . . .

3          1/3       2/3       3/3       4/3 . . .

A counting rule needs to be an algorithm that determines which is the next number to be counted and that counts each element in a set once and only once. The counting rule here is:

1. Begin at 0.

2. Count one unit to the right on the x-axis.

3. Count diagonally down and to the left until reaching the y-axis.

4. Count one unit down along the y-axis.

5. Count diagonally up and to the right until reaching the x-axis.

6. Repeat steps 2 to 5 ad infinitum.

This zigzag path ensures that every combination of two natural numbers will be counted once and only once; hence the number of all fractions also equals א0. If one replaces the above grid with a multiplication table, then counting will show that the number of all composite numbers is also equal to א0. Further, since the table multiplies the א0 set along the x-axis by the א0 set down the y-axis, we can see that aleph null squared equals aleph null..

An equivalent notation, which will quickly become useful, is to use paired numbers. That is, just as binary notation produces a sequence of countable numbers: 0, 1, 10, 11, 100, 101, 110, 111, etc., so can a dyadic notation. That is, (0,0), (0,1), (1,0), (1,1), (0,2), (1,2), (2,0), (2,1), (2,2), (0,3), etc. Every possible pair of numbers will be generated.

The points in a three-dimensional array can also be counted. One can visualize adding a z-axis and using a counting rule that works like a zigzag spiral: (0,0,0,), (0,0,1), (0,1,0), (0,1,1), (1,0,1), (1,1,0), (1,1,1), (0,0,2), etc. A correctly defined counting rule will always establish which combination comes next, will generate every possible combination of three numbers, and will ensure that each combination is counted once and only once. Since each axis provides an א0 set, aleph null cubed equals aleph null.

One can advance to a four-dimensional array. Humans cannot simply visualize that, but we don’t need to. Using a counting rule on quadruplet sets, (0,0,0,0), will produce the same result. Likewise for five dimensions and on up. Hence א0 raised to any finite power still equals א0.

Then is every definable unending set of numbers equal to aleph null? No—and this is where the train of thought gets more interesting. The number of all irrational numbers is larger than א0. Here’s why. An irrational number is a nonrepeating decimal number, like π = 3.14159…. Take any two decimal numbers that are the same out to some position, say, 0.2345 and 09.2346. Is the latter the next number after the former? No, because one can generate 0.23455, which is between them. If one subdivides an inch, there is no end to the subdividing: one can write an infinite number of decimals within any distance, no matter how small. Since there is no way to determine a “next” number, there can be no counting rule. Kantor called this “number of all irrational numbers” the “number of the continuum,” or Aleph sub c. Hence there are at least two different transfinite numbers, two different infinities, even in this simple mathematical model.

Just for thoroughness, note that whereas א0 raised to any finite power still equals א0, א0 raised to the א0 power is larger than א0, again because there is no way to determine a “next” number and therefore there can be no counting rule. The last I heard, no one has yet been able to decide whether aleph sub c is the same as א0 raised to the aleph null power..

Wrapping up this part of our program, I will note that, according to Hans Hahn (in his essay “Infinity” in James R. Newman, ed., The World of Mathematics [Simon and Schuster, 1956], III, 1598), Kantor also created an algorithm that will generate an unending sequence of different transfinite numbers, that is, an infinity of infinities. I have to take that on faith. The math is beyond me. I think it’s an argument somewhat similar to Gödel’s Proof.

In our next episode, we will begin to deconstruct the theology of infinity.





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