Is Math Broken? The Problem of Infinity.

I find infinity fascinating. It’s such a weird concept. Wikipedia says “Infinity ( $\infty$ ) is the notion of boundlessness in mathematics, philosophy, and theology.”

There are numerous paradoxes associated with the infinite. Take Ian Stuart’s Hyperwebster Dictionary. This paradox shows how subtracting infinity from infinity equals… infinity!

The Hyperwebster is a dictionary of all possible letters and combinations of letters using the 26 letters of the English alphabet. It starts out like this:

A, AA, AAA….
AB, ABA, ABAA, ABAAA…
AC, ACA, ACAA…AZ, AZA…
B, BA, BAA, BAAA, … BAB, BABB, BABBB…
C, CA…
ZA, ZAA…
ZZZZZZZZZZZZZZZZZZZZZZ…..

Now this is quite a large dictionary, so the publishing company decides to break it into 26 volumes, one for each letter. Since the company is concerned about ink, they decide to also drop the first letter in each volume, since everyone knows that in the A volume each entry starts with the letter A, etc. So, their first volume looks like this:

A, AA, AAA….
AB, ABA, ABAA, ABAAA…
AC, ACA, ACAA…AZ, AZA…
B, BA, BAA, BAAA, … BAB, BABB, BABBB…
C, CA…
ZA, ZAA…
ZZZZZZZZZZZZZZZZZZZZZZ…..

But wait, dropping the first A does nothing. No, it does more than nothing, it means the first volume now contains the entire dictionary. How can subtracting an infinite number of letters not change anything? Like I said infinity is weird.