Outside of mathematics, it’s considered a problem when a theory generates paradoxes. But within mathematics, some paradoxes are accepted and celebrated—as if they are special insights into the magical and mysterious world of math.
The concept of the infinite totality is the #1 generator of mathematical paradoxes, and over the course of a few centuries, instead of rejecting the concept as flawed, mathematicians enshrined it at the foundation of their theories. Today, most people cannot imagine that this concept is flawed—in fact, given the intellectual social hierarchy, people are more willing to accept patent absurdities (i.e. real paradoxes) than to entertain the idea that mathematicians are wrong.
Endless criticism can be directed at the concept of the completed infinity, but rather than engage in philosophical analysis, I thought it would be helpful to list a few uncontroversial paradoxes—that is, those paradoxes that are actually accepted by the mainstream.
Don’t take my word for it. The mathematicians speak for themselves:
Perhaps the most famous of the paradoxes of infinity, Hilbert’s Hotel is a thought experiment where a hotel has an infinite number of rooms, all of which are occupied. Yet, it can still accommodate additional guests, simply by moving them around in a clever way. In fact, even if every room is full, the hotel can accommodate an infinite number of new guests.
If that doesn’t make sense, then good for you. You have not lost your mind yet, unlike the producers of this informative TED video with 23M views:
2. Gabriel’s Horn (the Painter’s Paradox)
Gabriel’s Horn is an object with an infinite surface area yet only a finite volume, which leads to the famous Painter’s paradox where these two propositions are simultaneously true:
(A) The horn can hold a finite amount of paint within it (exactly π units of paint). Yet,
(B) The surface of the horn cannot be painted with a finite amount of paint
Listen to this video explaining it. Or skip to the 16 minute mark to see this mathematician embrace the absurdity by saying, “That’s the paradox! I tried to think about that and was like [mind blown].”
3. The Banach-Tarski Paradox (BTP)
The BTP is supposed to demonstrate that it’s possible to deconstruct a sphere into a finite number of parts, and simply by re-arranging the parts, you can end up with a duplicate sphere of the same size. The sphere gets doubled.
(I know, I know, your textbook told you that the BTP follows from the axiom of choice, not the axiom of infinity. But infinity does still make its way into the paradox, because it requires the spheres to be broken down into a finite number of sets composed of an infinite number of points.
If you want to attribute the paradox to Choice instead, that’s fine, but then it becomes a demonstration how concepts other than infinity can draw mathematicians into self-evident absurdity.)
This explainer video by the popular VSauce has a whopping 40M views:
For more than a century, mathematicians have gotten away with accepting paradoxes, celebrating them, and even using them as demonstrations of their formidable intellect. But this situation will not last forever. As we emerge from our present dark age, cherished assumptions in math will be revisited, revised, and the discipline will be put on discrete foundations. And lots of people will rightfully be embarrassed.
