## Increasing vs. Unbounded

This post is pure digression from the Galois theory arc, but I think it’s worth making.  There are some fundamental confusions which seem to be endemic among non-mathematicians.  I was reminded of this one as part of the geometry course I am teaching, in a discussion of Aristotle’s axiom.  I thought to myself then that this conflation of ideas had nothing to do with geometry, was much more primitive than that, and I was surprised by how hard I had to work to get my students to see the light.

The problem, I think, is one of language.  The imprecision of language because imprecision of thought.  In this case, the key phrase is “bigger and bigger and bigger”.  Suppose there is some quantity which changes over time, and someone says it gets bigger and bigger and bigger.  What would that mean?  There are two primary interpretations.

1. the quantity is larger each moment than it was in the past; i.e., the quantity is increasing.
2. the quantity will eventually get larger than 1000, larger than 1000000, larger than any quantity you can name; i.e., the quantity is unbounded.

Well, you might think that these are pretty much the same thing.  If something is always increasing, won’t it have to eventually be huge?  The answer is an emphatic no.

Consider the following graph of the function $y=f(x)=\frac{e^x-1}{e^x+1}$ (made on wolframalpha).  As you move the right, the graph gets higher and higher.  This function is increasing.  But it never reaches its asymptote at $y=1$, and it certain never achieves values as large as 1000.

Perhaps you find this unpersuasive, because I gave you a graph and a formula.  Perhaps you feel that I’m cheating by resorting to esoteric mathematical stuff.

So I will tell you a story, the story I told my class which made the penny drop.  Imagine a person, my personal grader, with a very simple life.  Each day my grader gets up, picks up the stack of grading for the day, grades it until it’s done, and then plays video games until the day ends.  Every day he gets the same amount of grading to do, and it’s always the same monotonous task.  The first day he doesn’t finish until 7 p.m., and he gets 5 hours of video games.  But the grader gets better and better at grading, a little more efficient every day.  So each day he finishes a bit sooner.  Then his daily video game time is “getting bigger and bigger” in the sense that it is increasing.  If you really believe that increasing implies unbounded, then you must also believe that, eventually, he’ll be playing video games for hundreds of hours, hundreds of years, hundreds of centuries, every day.