An infinite dictionary: countably infinite or uncountably infinite? This question concerns Ian Stewart's "Hyperwebster", an uncountable dictionary. 

Say a publishing company wants to publish every possible permutation (of any length) of the characters A-Z. The dictionary might look 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 publishing company realizes that the dictionary can be reorganized into 26 volumes, with each volume corresponding to one of the 26 characters:


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, ...


The company can save some ink by dropping the first letter, since it can be inferred from each of the 26 volumes.


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, ...


Since each volume is now identical, the company decides to publish only the first volume, which is the same as the list we started with. 

In the linked article, the author remarks that this is an analogy of the real line. Can someone explain how this dictionary is uncountably infinite? Is it because of the fact that we can keep repeating the above steps, subdividing our dictionary infinitely many times, each time arriving at a set the same size as the one we started with?
The dictionary is countable (as words have finite length). If there is an analogy with the real line it does not extend to the cardinality.
Try matching the letters with countable infinity. 
A, AA, AAA, AAAA, AAAAA... 
1, 2,  3,   4,    5...
This will extend to an infinite amount of As using up every countably infinite number.
By the time we get to AB, ABAA, ABAA... we will have already run out of natural numbers because they were all used up in the set of infinite As. 
Because we cannot match each word to a corresponding natural number. The amount of words must be unaccountably infinite.