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.