Hyperwebster - an uncountable dictionary

Introduction

There are an infinite number of points on the real line. In fact, there are more points on any segment of the real line, no matter how small, than all of natural numbers combined. This was proved by Cantor using the diagonal argument.

To get an idea of the number of points on a real line, Dr. Ian Stewart gave the following construction of an infinite dictionary, the "hyperwebster".

Construction of the Hyperwebster

An enterprising publishing company decides to print a book containing all the words that can possibly created from the English alphabet A-Z. Since it is really a collection of letters of the alphabet in any order, of any length, we will see:

• some non-sensical words like "XWPBQI" and "NKPMZ",
• some real words like "SUN" and "MOON" and
• even some composite words like "SUNMOON" and "MOONSUN".

Since not all such words have meaning, the publishing company decides to include only the words in the book, without their associated meaning, if any. 

So the "Hyperwebster" book looks like this:

A, AA, AAA, ..., AB, ABA, ABAA, ..., AC, ..., AZ, AZA, ...

B, BA, BAA, ..., BB, BBA, BBAA, ..., BC, ..., BZ, BZA, ...

C, CA, CAA, ..., CB, CBA, CBAA, ..., CC, ..., CZ, CZA, ...

Z, ZA, ZAA, ..., ZB, ZBA, ZBAA, ..., ZC, ..., ZZ, ZZA, ...

The staff at the publishing company realizes that it can partition the words into 26 volumes, one for each letter of the alphabet. So the Hyperwebster now looks like the following:

Volume A: A, AA, AAA, ..., AB, ABA, ABAA, ..., AC, ..., AZ, AZA, ...

Volume B: B, BA, BAA, ..., BB, BBA, BBAA, ..., BC, ..., BZ, BZA, ...

Volume C: C, CA, CAA, ..., CB, CBA, CBAA, ..., CC, ..., CZ, CZA, ...

Volume Z: Z, ZA, ZAA, ..., ZB, ZBA, ZBAA, ..., ZC, ..., ZZ, ZZA, ...

Next, the staff realizes that all words in volume A start with the letter A, all words in volume B start with the letter B and so on. This means that the first letter in each word can be inferred from its volume and hence, the first letter can be dropped. Excellent! The publishing company just saved some ink by not printing an infinite number of letters.

The new volumes now look like this:

Volume A: A, AA, AAA, ..., B, BA, BAA, ..., C, ..., Z, ZA, ...

Volume B: A, AA, AAA, ..., B, BA, BAA, ..., C, ..., Z, ZA, ...

Volume C: A, AA, AAA, ..., B, BA, BAA, ..., C, ..., Z, ZA, ...

Volume Z: A, AA, AAA, ..., B, BA, BAA, ..., C, ..., Z, ZA, ...

The staff realizes that each volume now looks identical, except for the name of the volume. Why would anyone buy 26 identical copies of the same content? So, the decision is made to publish a single volume called "Hyperwebster", which looks like the following:

New hyperwebster:

A, AA, AAA, ..., B, BA, BAA, ..., C, ..., Z, ZA, ...

This turns out to be identical to the original hyperwebster that they started out with.

Original hyperwebster:

A, AA, AAA, ..., AB, ABA, ABAA, ..., AC, ..., AZ, AZA, ...

B, BA, BAA, ..., BB, BBA, BBAA, ..., BC, ..., BZ, BZA, ...

C, CA, CAA, ..., CB, CBA, CBAA, ..., CC, ..., CZ, CZA, ...

Z, ZA, ZAA, ..., ZB, ZBA, ZBAA, ..., ZC, ..., ZZ, ZZA, ...

The staff realizes that:

1. the original volume can be partitioned into 26 different volumes,
2. the first letter in each volume can be dropped, making each volume identical,
3. and each volume now is really identical to the original volume
4. and steps 1-3 can be applied ad infinitum.

The publishing company wisely abandons publishing the hyperwebster, even though each execution of steps 1-4 represent an infinite amount of savings!

Moral of the story

The content of the hyperwebster is equivalent to points on a real line (replace A-Z above with 0-9 and observe that it generates all real numbers). Any subset of the real line can be chopped up into infinitely many parts, each of which has the same number of points as the original. Each of the parts in turn can be chopped up into infinitely many subparts, each having the same number of points as the original, ad infinitum. Yeah, that's a lot of points! Continuum hypothesis states that the number of such points is aleph-1.

References

• Leonard M. Wapner, "The Pea and the Sun", 2005.