Two infinities and two more infinities
If you are interested in math and science, you probably have heard a fair bit about infinities. There are countable and uncountable infinities with different cardinalities of their sets, which is usually phrased as “some infinities are bigger than others”. They come with terms like alef (\(\aleph\)) and beth (
I have seen some videos of physicists jumping into this popularization fray, parroting what the mathematicians say. But their participation betrays the fact that they don’t really understand which infinities are relevant in physics.
The only two types of infinity a physicist needs to know are actual infinity and potential infinity. One is to be used, and the other is to be avoided.
Why just those two for physicists? It’s because infinity is a concept and not a number. We are what I would call numbers people. We live to push numbers around all day long. And you can’t put a concept in place of a number. If you do, you get contradictions and absurdities.
Actual infinity
Let’s start with the troublesome one: actual infinity. This means I must make a cursory mention of set theory.
In set theory (a branch of mathematics), they consider collections of objects. Roughly speaking, a set is a bunch of objects you draw a circle around. (Technically, it’s curly braces {} around them.) These objects could refer to real-life things, like all the wild ocelots in the US. The circled objects can also be mathematical objects, like the set of integers greater than 41 and less than 43. Those are both finite sets.
Mathematicians have a concept of infinite sets, too. They postulate that it is possible to instantiate an infinite number of mathematical objects into a set. That is, they have a collection that they can draw a conceptual circle around and say, “There’s an infinity of math things inside.” It is a set of things that is larger than any finite set of things. That is actual infinity. Its symbol is $$\Large \infty$$And it has no place in physics. A physicist needs to know this concept so that they stay away from it.
Our two infinities will also give us two infinitesimals.
Actual infinity gets you what I call the mathematician’s infinitesimal. The idea originated from trying to divide a continuum into infinite parts. I will represent that as dividing a number by infinity:
$$ \frac{1}{\infty}$$
Nowadays, a mathematician’s infinitesimal is defined as a number greater than zero and less than the smallest positive real number. This infinitesimal is an absurdity in standard math. It’s easy to see that it can’t exist. Say you had the smallest positive real number. Now, divide that by two. You now have an even smaller positive real number. But we just said you already have the smallest positive real number. How can there be something smaller than the smallest number? We’ve come up to a contradiction. So, it is illogical to say that the mathematician’s infinitesimal can exist.
Please note that I am not saying mathematicians believe that this infinitesimal exists. I gave it this name because this idea is what they mean when they use the word infinitesimal and I need to distinguish it from another infinitesimal.
Potential infinity
Potential infinity, on the other hand, is not a grouping of things. It is a direction and a process, an endless and limitless process at that. You can think of it in terms of the horizon. I can point you to the horizon, and you can walk toward it. But you’ll never get there.
Physicists, engineers, mathematicians, chemists, and all other STEM majors use potential infinity. It is the infinity of calculus.
It is symbolized like this: $$x\rightarrow \infty$$ This can be read as “\(x\) approaches \(\infty\)”. Remember, though, it never gets there. It can be used in expressions like this:
$$\frac{1}{x}\rightarrow 0, \quad \text{as}\ {x\to\infty}$$
This can be read as “one over \(x\) approaches zero, as \(x\) approaches infinity.” However, our sloppiness with language has caused us to lose mathematical understanding over the generations. As an example of sloppiness, we might write the following:
$$\frac{1}{x}=0, \quad \text{as}\ {x\to\infty}$$
Which is wrong. The function never equals zero because there is no number you can put in place of \(x\) to make the function zero. A physicist will often say something like, “The function is zero when \(x\) is at infinity.”
The loss of mathematical understanding is not without consequences. Prof. Stephon Alexander, a cosmologist at Brown University, said on StarTalk,
Some people say you have to accept the infinity.
He was referring to some astrophysicists’ and cosmologists’ opinions on whether there is an infinite density singularity inside some black holes. For decades, a fraction of theoretical physicists have been saying there is an infinitely dense mass at the center of some black holes.
To get to that unphysical absurdity, one must not be able to distinguish between an actual realization of infinity and a limitless process. In mathematics, you can jump around freely to various solutions/frameworks without concern for physical processes. On the other hand, if you want to model the actual universe, you best start with a physically reasonable scenario, end with a physically reasonable scenario, and have a smooth process for getting from one to the other. A physicist shouldn’t be like a child who is challenged to count to infinity by a friend: “One, two, three … infinity!”
This is not some fringe group of physicists that I am calling out, nor is my example of childish behavior misplaced. Roy Kerr (of the Kerr metric/black hole fame) published a pre-print manuscript eight months ago titled Do Black Holes have Singularities? The abstract reads, in part:
There is no proof that black holes contain singularities when they are generated by real physical bodies. Roger Penrose[1] claimed sixty years ago that trapped surfaces inevitably lead to light rays of finite affine length (FALL’s). Penrose and Stephen Hawking[2] then asserted that these must end in actual singularities. When they could not prove this they decreed it to be self evident. …
In the third and fourth sentences, Kerr says that Penrose and Hawking did a “One, two, three, … infinity!”
The physicists’ infinitesimal
Not understanding a limitless process also leads to a misunderstanding of another quantity that I will call the physicist’s infinitesimal (technically, it’s called a differential). I apologize to the chemists, engineers, computer scientists, etc. for the biased name. I know you all use it, too. I say it’s the physicist’s infinitesimal because calculus was invented for physics, and the founder of calculus is our guy.
This infinitesimal is symbolized by the letter “d” followed by a variable: \(\mathrm{d}x\). Mathematically, I describe it like this:
$$\mathrm{d}x = \Delta x \to 0$$ The symbol \(\Delta x\) by itself is a variable and is a finite distance or quantity. By adding the symbols “\(\to 0\)”, we are saying that \(\Delta x\) gets arbitrarily small but never gets to zero. This process never reaches the limit of zero. The physicist’s infinitesimal is a finite but non-zero quantity that is as small as needed for the problem at hand.
To avoid some tsk-tsk comments from mathematical purists, I must mention that this is not the exact definition. However, it is a good definition because it gets my point across.
Easy to understand, right? Nope.
Quantum mechanics is infected, too
The confusion over infinities and infinitesimals happens in quantum mechanics, too. In my book Physicists at Fault, I point out where physicists have arrived at absurd conclusions that arise from the confusion over these topics. I make the most explicit reference to their absurd claims in Misconceptions about the electron in particular, Math vs Measurement, and the chapter on Schrödinger’s equation As a wave equation. I think that any physicist who says that the electron is a geometric point is a person who doesn’t understand that actual infinity is not a number and that the mathematician’s infinitesimal is logically impossible.
Finally, I must end by saying my book doesn’t use every math and physics term perfectly. At times, I knowingly misuse technical terminology in minor but permissible ways for educational purposes. My book is not an encyclopedia or a textbook, so I omit the full scope of a concept or neglect special caveats. I certainly don’t give any topic a fully rigorous mathematical treatment.