mbm9123 – Proofs in the Pudding

The Infiniteness of Words

May 31, 2023mbm9123 2 Comments

Maya Mubayi

Countable, uncountable, and infinite sets can be found everywhere, from pure mathematical cases to sets we find in everyday life. We know that sets of numbers, such as $\mathbb{R}$ , $\mathbb{N}$ , $\mathbb{Z}$ , and many more can be classified as countable, uncountable, and infinite. Similarly, you can think of a tree as an infinite or finite set of branches, or even your own wardrobe as a finite set of pant-shirt-shoe combinations. What about the words in the English language? The number of possible English sentences? The number of possible books? Are those finite sets? Infinite sets? Uncountable sets? I will explore these questions further in this blog.

Why Are the Set of English Words Finite?

First define $E$ as the set of all English words. Since we are dealing with words, we can assume that there is in fact a longest word in the English language. Let the length of this word be $n$ for $n \in \mathbb{N}$ . If $E_{1}$ is the set of all English words of length 1, $E_{2}$ is the set of all English words on length 2, and so on, we know that $E_{1} \cup E_{2} \cup$ … $\cup E_{n} = E$ . Since there are at most 26 options for any letter, for each $E_{i} \subseteq E$ for $i \le n$ . This means that $E$ is composed of a finite union of finite sets.… Read the rest