# To Infinity…and Beyond!

In researching Gödel’s Incompleteness Theorem, I stumbled upon an article that stated no one has proven a line can extend infinitely in both directions. This is shocking, if it’s true, and after a quick Google search, I couldn’t seem to find anything that contradicts the claim. So, in the spirit of intellectual adventure, I’ll offer a fun proof-esque idea here.

Consider a line segment of length $\ell$ that is measured in some standard unit of distance/length (e.g., inches, miles, nanometers, etc.). We convert the length of $\ell$—whatever units and length we’ve chosen (say, 0.5298 meters)—into a fictitious unit of measurement we’ll call Hoppes (hpe) [pronounced HOP-ease]. So, now, one should consider the length of $\ell$ to be 2 hpe such that $\ell$/2 = 1 hpe. We then add some fraction (of the length of) $\ell$ to (both ends of) itself, and let’s say the fraction of $\ell$ we’ll use, call it $a$, is 3$\ell$/4, which equals 3/2 hpe. The process by which we will add $a$ to $\ell$ will be governed by the following geometric series:

$s_n(a) = 1+a+a^2+a^3+\dots+a^{n-1} = (1-a^n)(1-a)^{-1}=\frac{a^n-1}{a-1}$.

Let us add the terms of $s_n(a)$ to both sides of $\ell$; first, we add 1 hpe to both sides ($\ell=4$ hpe), then 3/2 hpe ($\ell=7$ hpe, then 9/4 hpe ($\ell=23/2$ hpe)and so forth. If we keep adding to $\ell$ units of hpe based on the series $s_n(a)$, then we’re guaranteed a line that extends infinitely in both directions because $\lim_{n\rightarrow\infty} (a^{n}-1)(a-1)^{-1} = \infty$ when $\vert a\vert \geq 1$.

Now, suppose we assume it is impossible to extend our line segment infinitely in both directions. Then $s_n(a)$ must converge to $(1-a)^{-1}$, giving us a total length of $2+(1-a)^{-1}$ hpe for $\ell$, because $\lim_{n\rightarrow\infty} 1-a^{n}=1$, which is only possible when $\vert a\vert < 1$. (We cannot have a negative length, so $a\in \text{R}^+_0$.) But this contradicts our $\vert a\vert$ value of 3/2 hpe above, which means the series $s_n(a)$ is divergent. Q.E.D.

N.B. Some might raise the “problem” of an infinite number of discrete points that composes a line (segment), recalling the philosophical thorniness of Zeno’s (dichotomy) paradox; this is resolved, however, by similarly invoking the concept of limits (and is confirmed by our experience of traversing complete distances!):

$\sum_{i=1}^{\infty} (1/2)^i=\frac{1}{2}\sum_{i=0}^{\infty} (1/2)^i=\frac{1}{2} s_n (\frac{1}{2})=\frac{1}{2}\Big( 1+\frac{1}{2}+(\frac{1}{2})^2+\cdots\Big)=\frac{1}{2}\Big(\frac{1}{1-\frac{1}{2}}\Big) = 1$,

a single unit we can set equal to our initial line segment $\ell$ with length 2 hpe.

Special thanks to my great friend, Tim Hoppe, for giving me permission to use his name as an abstract unit of measurement.