Prove that not every real number can be named. A "name" is a finite string consisting only of the $26$ letters of the English alphabet.
The inspiration for this question was this meme.
Let $\mathbb{N}_{\leq 26}=\{1,2,\ldots,26\}.$ Let $\mathbb{A}=\{a,b,\ldots,z\}.$ An obvious bijection exists from $\mathbb{A}$ to $\mathbb{N}_{\leq 26}.$ Let $\mathbb{S}$ be the set of all names that can be formed using the specified rules. Each name is a finite string of letters. This means that each name can be seen as a finite sequence of numbers less than or equal to $26.$ This is because of the bijection between $\mathbb{A}$ and $\mathbb{N}_{\leq 26}.$ For example, the name "twelve" can be seen as the finite sequence $\langle20,\,23,\,5,\,12,\,22,5\rangle.$ Also, to each finite sequence of natural numbers less than or equal to $26,$ there is an associated name. This means that $|\mathbb{S}|=|F(\mathbb{N}_{\leq 26})|,$ where $F(\mathbb{N}_{\leq 26})$ is the set of all finite sequences of natural numbers less than or equal to $26.$ But, $F(\mathbb{N}_{\leq 26})$ can be injectively mapped to $F(\mathbb{N}).$ Just map any element of $F(\mathbb{N}_{\leq 26})$ to itself, but in $F(\mathbb{N}).$ This can be done since $F(\mathbb{N}_{\leq 26})\subset F(\mathbb{N}).$ This gives $|F(\mathbb{N}_{\leq 26})|≤|F(\mathbb{N})|=|\mathbb{N}|.$ So, $|\mathbb{S}|≤|\mathbb{N}|.$ This means that there is no surjection from $\mathbb{S}$ to $\mathbb{R}.$
Is this proof fine?
Words are sequences of agreed upon symbols, so there are a finite number of symbols and signifiers for words, the best you can do with a finite number of symbols to make a finite expression for each number is to express an opposite set of natural numbers, such as the set of rational numbers or roots of natural numbers Or something like that, since the real numbers are greater than the natural numbers, you cannot represent them by letters, because this means that there is a correspondence between the natural numbers and the set of real numbers, and the existence of such a correspondence is impossible, you can see Cantor's diagonal argument, for example, to prove that
https://en.m.wikipedia.org/wiki/Cantor%27s_diagonal_argument
You can repeat the same argument by using letters instead of numbers to make proof of your requirements
You may also be interested in reading paradox berry
https://en.m.wikipedia.org/wiki/Berry_paradox