There Is More than One Infinity

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.





  • Robert Mathiesen

    Mathematical theology: I love it!

    Kantor was a brilliant. brilliant mathematician. (I have heard, though I haven’t tried to verify it, that he was also very interested in Kabbalah, which led him to use aleph as his mathematical symbol of choice for his several transfinite numbers.)

    As for Gödel you can probably use his incompletenss theorem to build a mathematical argument, maybe even a mathematical proof, that the “Matrix hypothesis” is false and the part of our real world that is mathematics or can be put in correspondence with mathematics can not have been generated algorithmically, that is, by any sort of simulation program that has been written along the lines of our own computer programs.

    There is an old story, possibly apocryphal, that Einstein once explained why he had moved to the Institute for Advanced Study” at Princeton: “because there I can talk with Gödel.” That was all the reason he needed.

    • aidanakelly

      Well, this is cool. I wasn’t sure I’d get any comments on this at all. But this is just a prolegomenon. Both Joseph Weizenbaum and Roger Penrose have proved that the AI goal of replicating human thinking is inherently impossible, But the AI folks, who seem to think a lot like “atheistic materialists,” continue trying to build the equivalent of a perpetual motion machine.

    • Dave Burwasser

      Robert, Goedel’s proof operates at the level of propositions about mathematics, not mathematics itself. It says, concisely, that with an axiom set rich enough to derive mathematics, there will always be true statements that cannot be proved. The matix method of counting all rational fractions (and showing them to be of the same level of infinity as the integers) is valid.
      Aiden, you have a critical typo. You have 09.2346 where you meant 0.2346. A nice run through the groves of transfinites, otherwise.

      • Robert Mathiesen

        I wasn’t arguing against the matrix method of counting all rational fractions, nor about mathematics in itself. I was chasing a different rabbit, which was the following:

        Algorithms and methods of mathematical proofs have similar logical structures in certain ways. So Gödel’s valid result showing that certain true statements in mathematics escape any possible mathematical proof under a sufficiently rich set of axioms might well be generalizable to an equally valid result about algorithms, namely, that certain true statements about the world of matter and energy might escape derivation from a set of algorithms (or a computer program embodying them) sufficiently rich to derive most true statements about that world of matter and energy. If so, would that not bear on the “Matrix” hypothesis (that the apparent world of matter and energy in which we live is a computer simulation)? Or have I missed some key point?

  • P. Sufenas Virius Lupus

    This is fascinating…
    However, I have a kind of caveat to add. I don’t think the problem is so much that “theorized monotheistic god is infinite, therefore polytheistic gods must be finite” is the reality in many of these cases (except in the “teenage rebellion, our gods can’t be anything that the monotheistic gods are” schools of theology); what is more often the reality is that “our gods don’t have to be infinite to be gods.” Sure, an Abraxas here and there can be a lot of fun, and very useful; but, just because Zeus isn’t “the All” and “the Infinite,” and in fact has a very delineated role within his cosmology, doesn’t mean he’s any less valuable for being finite.
    But, as you’ve described above, there’s an infinity of divisions between each whole number, and thus even apparently limited and finite gods are not necessarily as finite as they may seem…
    (Which then reminds me what Brian Greene and some physicists have said about parallel universes and such: a number of them may appear finite from the outside, but infinite from within them, which is as good a description as any of what I just tried to express with the infinite divisions metaphor above…!?!)

    • Scott Martin

      Or the observation that there are finite-but-unbounded entities: the surface of a sphere is a finite area, but travel upon it is unconstrained by “edges,” and so one only gets a “proper” sense of its finitude when viewing it from outside. Hawking has deployed this analogy to explain a finite-but-unbounded model of the universe, with further reference to a balloon’s surface to demonstrate how that surface may expand (and indeed may appear to be expanding “out from” *any* arbitrary point on the surface).

  • Soliwo

    I do not understand math and I do not see the gods as being mathematical constructions, so I look forward to your next article. I wonder,why would the gods be infinite, or why would you want them to be infinite?

  • Soliwo

    “I have unfortunately seen some people argue that, “Since the Christian God is infinite, our Pagan Gods must be finite.”

    I think most ‘people’ do not argue this. Many more argue, we should not assume the gods are infinite just because the Christians claim their god is infinite. And there are other better though-out arguments for the gods being finite. It would be more interesting to see you arguing your case (the gods are infinite), without you using the above statement as as a straw man, but instead addressing some of these.

    • Robert Mathiesen

      In my experience, it is generally Christians who make the claim that Aidan has rightly destroyed here.

      • Soliwo

        But why would Christians argue that our gods must be infinite? In my experience they either believe they are non-existent (and thus very much finite) or some demons (and thus finite, for God is more powerful).

        • Robert Mathiesen

          No, the Christians’ argument is that other Gods, including Pagan Gods, are finite, and thus they are not really Gods at all, but some order of lesser spirits — and rebel spirits at that. I didn’t read Aidan himself as arguing for finite Gods

    • Soliwo

      Sorry, this came out really harsh. But do you I really do doubt many Pagans will use the exact phrasing used above. And though I am open to other infinities (I am curious about your next piece), I do think that the tendency for Pagans to imagine their gods to be infinite, has to do with the Christian heritage of the Western world, and less with mathematics.

  • Omorka

    While this is all certainly mathematically true, what is more interesting to me is that there are an infinite number of defined number sets – some of them quite limited in scope indeed – that still have aleph-null elements, or even c elements in some cases. The set of all prime numbers is one of these, as is the set of all powers of two. Those both have aleph-null elements, and yet their common subset contains but a single integer, {2}. I think of the gods as being like these sets, infinite in scope in certain respects and yet well-bounded in others, rather than trying to cover everything all at once.

  • Ian

    Man, the 15 year old in me is totally grooving on the Aleph talk; thanks for that.

    I have more often heard the theological distinction made, though, not between the finite and infinite but the infinite and the eternal; the Infinite is a property of material and temporal existence while the Eternal transcends the world in which infinity operates, generating it without being subject to it. What you do with set theory fits that vision of monotheism nicely.

    Just identify God with the uncountable infinite set–i.e., that which is greater than any specific infinite set defined by a pagan divinity. Especially since you seem to suggest pagan divinities be comparable but different infinite sets, you seem to suggest that they are distinct aleph null sets. Make God the uncountable set and you can start to talk about the idolatry of worshiping an aleph null set (e.g., the God that you can name is not the true God). Admittedly, that is likely to be a kinder, gentler, more gnostic (i.e., heretical) variety of the big Monotheisms, but still.

    Fwiw, I’m simpatico with that, but with your talk of ‘deconstructing’ infinity theology, I’m not sure that would really satisfy what you are looking for or not.

  • Marcílio Diniz da silva

    Mr. Kelly, hail.

    Can we translate this text to portuguese (with all the respective credits to you and the original post here) and publish it in a blog (this blog:

    Thanks in advance and waiting response.

  • Xen

    I will say, that math is one representation of reality, and as a representation is limited. The Real is actually much more strange.

    Thanks for a great example!

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