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.


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