The most common encounter with the concept of infinity is associated with the positive whole numbers 1, 2, 3, 4, 5, 6… , which go on without end. In ancient Greece, the celebrated mathematician Euclid famously proved (around 300 BCE) that there is even an infinite number of primes (numbers divisible only by 1 and themselves, such as 3, 7, 17, or 541). Not until the nineteenth century, however, did anyone find a way to actually rank infinities, and manipulate them in ways one would normally do with ordinary numbers. The person who in some sense “tamed” infinities, by demonstrating that they can be properly defined and arranged in a hierarchy in which each infinity is manifestly larger than the one below it, was Georg Cantor (1845–1918; Figure 1). Cantor first found a clever way to show that even though it seems that there are many more fractions, quantities such as 3/5, 9/11, or 241/509, than whole numbers, their infinities are actually of the same size! This sounds surprising, since clearly even just between 1 and 2 there are infinitely many fractions of the form p/q (where p and q are whole numbers). Yet, Cantor found a way to show that there is a one-to-one correspondence between the whole numbers and the fractions. In other words, the fractions (known as rational numbers) are definitely countable. Cantor’s recipe for how to do the counting is shown in Figure 2. First count all those fractions in which the nominator and denominator add up to 2, then those that add to 3, then to 4, and so on. Since this procedure clearly counts all the fractions, and each one is only counted once, you discover that the infinity of the fractions and of the whole numbers are of the same size. Cantor then proceeded to show that all the non-ending decimals are uncountable, meaning that the size of that infinity is larger than that of the whole numbers. In this way he constructed an endless hierarchy of infinities. For any given infinity size, one can construct an infinity of a bigger size. Cantor labeled the smallest infinity—that of the whole numbers—by the Hebrew letter aleph, to which he added the subscript zero, ℵ0. He then labeled all the larger infinities by increasing subscripts, ℵ0, ℵ1, ℵ2, ℵ3 …
Figure 1. The mathematician Georg Cantor. Credit: Wikimedia Commons.
Figure 2. A schematic demonstrating Cantor’s recipe for counting fractions. Cantor used this procedure to show that the size of the infinity represented by the fractions is equal to the size of the infinity represented by the whole numbers.
An intriguing question that arises is whether infinities are only a mathematical concept, or whether they can occur in physical reality. Interestingly, cosmology—the study of the universe as a whole—provides quite a few examples where in principle one could encounter infinity. First, there is the Big Bang itself—the singular event believed to have brought space, time, and our universe into existence. If the Big Bang (which we believe occurred some 13.8 billion years ago) should indeed be associated with a mathematical “singularity” (where one is essentially driven to divide by a size approaching zero), then quantities such as density (defined as mass per unit volume) would have had to be infinite.
Similarly, one could ask whether our universe is infinite in size, or whether it would exist for an infinite time into the future. Most physicists see singularities merely as an indication of the breakdown of the theory. In the case of the Big Bang, they point to the fact that we don’t yet have a quantum theory of gravity. Such a theory would unify our ideas about the largest cosmic scales (as expressed by Einstein’s General Relativity) with those on the subatomic scales (the quantum realm) and it may eliminate singularities and infinities.
We don’t actually know if our universe is infinite in size or not, but since our universe has a finite age, the (in principle) observable universe is definitely finite, with a radius of about 46 billion light-years. (One light year is the distance light travels in one year—about 6 trillion miles). Telescopes such as Hubble, and the upcoming James Webb Space Telescope have certainly expanded and will continue to expand our horizons far beyond what had been possible a century ago. The practical horizon of an optical or infrared telescope, no matter how powerful, is going to be limited by the fact that the universe was opaque to such radiation when it was younger than about 380,000 years. To probe the universe before that time, we would need different techniques such as gravitational waves or neutrinos.
Will our universe continue to exist for an infinite amount of time? We are not sure of that either. The mass of the recently discovered Higgs boson (Figure 3) suggests that the vacuum of our universe may be inherently unstable, meaning that at some point (tens of billions of years from now), our universe could be destroyed by a bubble “alternate” universe.
Figure 3. Representation of an event recorded at the Large Hadron Collider. Events like this led to the discovery of the Higgs boson. Credit: Wikimedia Commons.
To conclude, infinities do occur in a variety of physical theories. In some cases they may simply indicate that the existing theory is too naïve. In others, they may signal the existence of new physics. To some questions we may never know if the answer is infinity or not.