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 that is measured in some standard unit of distance/length (e.g., inches, miles, nanometers, etc.). We convert the length of —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 to be 2 *hpe *such that /2 = 1 *hpe*. We then add some fraction (of the length of) to (both ends of) itself, and let’s say the fraction of we’ll use, call it , is 3/4, which equals 3/2 *hpe*. The process by which we will add to will be governed by the following geometric series:* *

.

Let us add the terms of to both sides of ; first, we add 1 *hpe* to both sides ( *hpe*),* *then 3/2 *hpe *( *hpe*) *,* then 9/4 *hpe *( *hpe*)*, *and so forth. If we keep adding to units of *hpe* based on the series , then we’re guaranteed a line that extends infinitely in both directions because when .

Now, suppose we assume it is impossible to extend our line segment infinitely in both directions. Then must converge to , giving us a total length of *hpe *for , because , which is only possible when . (We cannot have a negative length, so .) But this contradicts our value of 3/2 *hpe *above, which means the series 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!):

,

a single unit we can set equal to our initial line segment 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.