I had thought of this idea last year, but never got around to writing it. In this blog, I show how humans are infinitely limited by the mode of language in transmission of ideas. I don’t quite remember how I got around to thinking about this, but here goes haha.
Every written language is described by a finite set of characters. Every spoken language requires a finite sequence of sounds. Every idea that is realized by humans and communicated through language has to be put into a sequence of letters / sequence of sounds. Call such a thought expressible.
I define languages to be made of fixed objects, i.e., there are no new characters. A reasonable contention would be: “this is BS? languages change all the time”. Well, yes. However, if they change, then I call the new object a new language. You can think of this as:
\[ \{\text{eng\_v1}, \text{eng\_v2}, \text{eng\_v3}\cdots\} \]
<fill in: why the set of all languages is countable>
Theorem
The set of all languages is countable.
Theorem
If the set of all possible spoken and written languages is countable then the set of expressible ideas is countable.
Proof.
Let \(S\) be the set of all languages. For a language, \(L \in S\), let \(\mathcal{C}(L)\) be the characters of \(L\). We understand that, \(|\mathcal{C}(L)|<\infty.\)
Define \(\mathcal{C}(S) = \cup_{L \in S}\{(L, x) : x\in \mathcal{C}(L) \}\), which is the set of all characters (signed with the language it belongs to). Since \(S\) is countable, \(\mathcal{C}(S)\) is countable.
Now, every expressible idea is a finite sequence of characters/phonemes like:
\[ \gamma_1 \gamma_2 \gamma_3 \gamma_4 \cdots \gamma_N \]
where \(\gamma_i \in \mathcal{C}(S)\).
Let \(\mathrm{I}\) be the set of all expressible ideas. We know that set of all finite sequences formed using countable characters is countable, and therefore: \(\mathrm{I}\) be countable. \(\blacksquare\)