There are several common proofs that cheat on the nature of infinity and end up making numbers falsely look equal to other numbers. In this page, I will address a quite simple proof that actually uses infinity. Here it is:
Because infinity is the largest number, adding one makes no difference. So here is the equation ∞ + 1 = ∞. Then you subtract infinity from both sides. After crossing out the infinites, you are left with 1 = 0. It’s that simple, yet leaves any unknowing thatalogist frustrated.
But if you’ve learned anything about thetalogy, you might remember that infinity minus infinity does not equal zero—at least not always. There are actually different values of infinity, so infinity minus infinity is not necessarily zero, it can also be positive or negative theta. We will never be able to more than comprehend the different values of infinity with our finite minds, so it must be grouped to a set of numbers. In turn, we also need a new set of numbers: rho. Rho (represented as Ρ) is equal to positive theta, negative theta, and zero. So, ∞ − ∞ = Ρ.
Now that that’s covered, we know that we actually can’t cancel infinity out from both sides of the original equation. Instead—after subtracting infinity from both sides—we are left with Ρ + 1 = Ρ. This looks much more correct. Any finite number, plus one, is a finite number.
But we are still left with one problem: rho plus one (or just rho) is only sometimes equal to rho, unlike what the equation suggests. But the truth is that it never was an equation. Infinity plus one is also only sometimes equal to infinity, so the original equation—more like semi-inequality—should actually look like: ∞ + 1 ⋛ ∞. Using the greater-than-less-than-or-equal-to sign causes it all to make sense again.
Now we can rest in peace, knowing of the ongoing exactness of algebraic thetalogy. That is, until someone brings up the equation of ∞ + ∞ = ∞, and subtracts infinity from both sides. 😉