How do you make sense of a claim that one infinity is different from, much less bigger than another? Well the trick is in two parts. First, it is taken for granted that when you talk about “one infinity,” you’re speaking about a collection of “objects” that cannot be counted by any finite number. (That’s a significant constraint, but we’ll come back to that.) Thus, we all know that the integers (0, 1, 2, 3, … [note the ellipsis]) have no largest number; they go on “forever,” they are infinite. Say then you have another collection, and you wish to test if it is infinite. The test is, can you correlate each element of this new collection with that of the integers, on a one for one basis, that is without any repeats or repetitions? If you can do that with the new collection, then it too is infinite. But what happens if there are more elements in this new collection than there are integers. Well, that is a really BIG infinite. And that’s the real numbers.
Funny fact, one of many, the fractions, the x/y numbers? Seems like there’s a lot of them, right? I mean, between 0 and 1 there are infinitely many fractions. And between any two fractions, you can always find infinitely many more. So that’s a BIG collection, right?
Yeah, not so much. Turns out there is a trick where you can correlate every fraction, one to one, with a unique integer. Those two collections are actually the same size. With infinities, appearances are almost always deceiving. Our brains aren’t born to think about such things. With training, discipline, practice, and whatever else keeps you going, one can educate both thought and intuition to follow these relational patterns. But it’s like hitting an overhead top-spin serve in tennis; you can’t even picture it until you’ve spent years with a mentor learning how.
So the collection of real numbers – you know, π, e, √2, crazies that can’t be written down in any finite list – is bigger than the integers and the fractions (which are the same size as one another). Once mathematicians, following geniuses like Georg Cantor, discovered these relationships, it became clear that there were infinities even bigger than these. (Also, hearkening back to my previous two posts, you might begin to sense how and why the solvable problems – which can be placed in a one-to-one correlation with the integers (or fractions) are so catastrophically overwhelmed by the unsolvable problems, which are one-to-one with the reals.)
This hierarchy of infinities made set theorists – the mathematicians who specialize in studying such things – very happy. But in the early ‘60’s Paul Cohen, who worked in an area of logic called “model theory,” demonstrated using his array of tools that one could prove there were infinitely many infinities in between the integers and the real numbers. For set theorists, this made no sense whatsoever, and for many years this led to many noses being thumbed at one another across this mathematical divide. (Set theorists tend to be very “ontological” in their view of things: their formulae and proofs are not merely logically coherent, they represent the REAL as such. Model theorists are content with logical coherence.) So the infinite, which was already bloody crazy, got crazier still.
But amidst all of these crazies, several of which are wildly incompatible with one another, there remains one commonality: they are all composed of points – “atoms,” if you will – that are the minimal units that make it possible to count, and thus determine, how big the collection is. But what if you’re dealing with a kind of “gunk” that has no minimal, nor any maximal, “unit”? A kind of “atomless gunk”?
Lest you accuse me of being silly, that is the technical name for a topic in “mereology,” the study of the logical relations of part-and-whole. What if there is no least part, no “atom” or basic element to the collection one is dealing with? To avoid prejudicing matters with biased terminology, this was called “atomless gunk,” and the name stuck (rather the way gunk always does.) This is a different kind of infinity, because it is no longer a matter of enumerating the size of the collection, but rather an operational infinity in which no amount of division or accretion can ever achieve a least or greatest element.ii Like the Koch snowflake, as one looks at smaller and smaller bits, one never encounters a smallest bit. Compare this to numbers where, say, the numbers “7” or “π.” These are absolutely atomic bits, “points” on the number line, in their respective infinite wholes. Which is to say, you can find a number smaller (or larger) than 7, but the point on the number line that corresponds to, or is named, “7” cannot be divided into smaller pieces. Similarly with other points on the integer, rational, or real number lines.
Now, with this little bit (so to speak) under our belts, what can be say about yet other things that are called infinite? For example, when someone says “God is infinite,” what (if anything) might that mean? Well, here we are dealing with the generic sloppiness of language, because until mathematicians and logicians got their prickly mitts on the topic, “infinite” was just the Latin version of “ἄπειρον,” and dogged by just as much vagueness.
The first stab at this might be to say that God is unbounded ( ἄπειρον), but this might not work. For consider the surface of a sphere: it is unbounded, and yet it is finite. It is not enough to say that God is of “unlimited extent,” for that is a spatial/temporal metaphor (which is already a mistake), and our friend the Koch snowflake (for example) is of unlimited extent, and yet it is strictly bounded. One can try to go the route of power – God is infinite in knowledge and ability – but that simply opens one to the problem of evil (why doesn’t God do something, when he is the author of it all?) and that is a rabbit hole that leads to no solutions.
I submit that this is the wrong direction to go with such a question; that is, size, power, BIG, is just wrong headed when one turns to theological issues. The Greeks thought the infinite, the infinite, the ἄπειρον, was evil because it exceeded the possibility of human interpretation. But perhaps that is not evil at all, any more than understanding how the real numbers exceed the rationals. There is a line of thought that says one’s reach ought to exceed one’s grasp. So even if the idea of God is beyond any of our concepts, we can still stretch ourselves by approaching – even if we never achieved – a concept of something we can never fully understand or represent.iii
– – – – – – – – – –
i Simply proving that √2 was not a fraction was a magnificent achievement. According to legend, the genius who produced this demonstration was rewarded for his effort by having a stone tied around his ankles and then tossed off a ship into the Mediterranean sea. The Greeks considered this kind of infinity a sign of profound evil.
ii Whitehead’s theory of extension – a central part of his entire philosophy of process – is built around a mereology of atomless gunk, a point that does not appear to be widely recognized in the scholarship.
iii This is very much of a piece with Plato’s dialog on love, the Symposium. The image is of a spiral that always approaches, but never reaches, the center point. Iris Murdoch, in her brilliant Metaphysics as a Guide to Morals, picked up this idea. She went well beyond just talking about it, and exemplified it as well in the very structure of her discussion.