Potential Infinity vs. Actual Infinity

What is infinity and does it even exist? In our everyday experience, we find only finite things. A basket of eggs contains only a fixed number of eggs and no more. Our bodies are composed of particles (molecules, atoms, protons, quarks, etc.). But whatever particles describe our make up, we find only a finite number. It may be billions or trillions or more, but it still doesn’t get close to infinity. Even the known universe is finite – it’s only so many light-years in diameter and contains only so many elementary particles.

How, then, does one even get close to infinity? People have long realized that there’s no biggest number because it’s always possible to add 1 to any number and get still a bigger number. So numbers themselves, taken collectively, are infinite. Any given number is finite, but the mere fact that numbers go on forever – that’s infinite.

But what sort of infinite is this? The ancient Greek philosopher Aristotle (384– 322 BC) proposed that there are two types of infinity, a potential and an actual infinity. In a potential infinity, one can keep adding or subdividing without end, but one never actually reaches infinity. In a sense, a potential infinity is an endless process that at any point along the way is finite. By contrast, in an actual infinity, the infinite is viewed as a completed totality. Aristotle rejected actual infinity, claiming that only potential infinity exists.

So what, you say? For all practical purposes, we get on quite well with quite a bit less than even a potential infinity. Take the world’s most powerful supercomputer, Japan’s K Computer, which runs at 10 petaflops, using 705,024 SPARC64 processing cores. There are absolute limits to what this machine can do in terms of storage, retrieval, and processing. It’s safe to say that 10^100 (i.e., the number 1 followed by 100 zeros, aka “google”) sets an absolute limit on the amount of processing steps this machine will ever do, on the length of the longest number it can compute, and on the amount of bytes available to the machine’s memory.

And yet, the infinite is not so readily cast aside for practical reasons. Modern mathematics is done almost entirely in terms of sets (recall the “New Math”). Set theory treats just about anything as a set (the only things that are not sets are things too big to be sets – more on that in another post). Now numbers are sets. For instance, 0 is the empty set (it contains zero items). The number 1 is also a set (it is the set that contains zero, and thus is a set with one item).

But all the numbers taken collectively (0, 1, 2, etc.) also form a set, known to mathematicians as the natural numbers and represented as {0,1,2,3,…}. Ah, but what’s that ellipsis, those three dots (i.e., …), doing there? Doesn’t that tell us that the natural numbers are really just a potential infinity? Mathematicians don’t treat the natural numbers as a potential infinity but as an actual infinity – a completed totality that includes all numbers 0, 1, 2, etc.

But what do mathematicians know anyway? Perhaps treating the natural numbers as an actual infinity is just a convenient way to think about numbers and do calculations. If people’s concerns about infinity were left simply at the level of mathematics and its scientific applications, the debate over potential and actual infinities would be moot. But it turns out that this debate spills over into other areas, notably theology. If God is real, is he an actual infinite or is he just a potential infinite? Most religious believers see God also as unchanging, so if God is real and infinite, he must be an actual infinity.

Now it’s interesting that Georg Cantor, who invented set theory over 100 years ago, did so in part for theological reasons, seeing the infinite sets he came up with as a reflection of the infinity of God. Others, however, not believing that God exists or thinking that the very concept of an actual infinity is incoherent, reject the actual infinity and thus view Cantor’s so-called actual infinities as simply a device for describing much more mundane and finite processes. Yet it is a device that every working mathematician uses. As the great mathematician David Hilbert put it, “No one will drive us from the paradise which Cantor created for us.”

The debate over potential and actual infinities has been ongoing for centuries, and this short post won’t resolve it. Nonetheless, it’s worth noting that Cantor’s work on set theory has showed that the concept of an infinite set makes mathematical sense and avoids contradiction. Certain paradoxes, such as that infinite sets can be put in one-to-one correspondence with proper subsets (e.g., there are as many even numbers as natural numbers: 0à0, 1à2, 2à4, 3à6, etc.), may fly in the face of common intuitions, but science confronts us with lots of things that are counterintuitive.

In any case, modern mathematics, especially in its wholesale incorporation of set theory, has given the single biggest boost to the view that the actual infinite exists. Not that this proves the actual infinite exists – the nature of existence itself (a field philosophers refer to as “ontology” – the study of being) is itself up for grabs. But the mere fact that treating mathematical entities as actual infinities has yielded incredibly fruitful mathematical insights (Cantor’s paradise) gives the actual infinite breathing room that it never had in the past.

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References:

Joseph Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (Princeton: Princeton University Press, 1990).

Michael Hallett, Cantorian Set Theory and Limitation of Size (Oxford: Oxford University Press, 1984).

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