In computer terms (which I'm more familiar with) this is the difference between expressing a program in some language vs evaluating that program.
Compare this with trying to define the "set of all possible concepts". What does that even look like? What is a concept? How can it be expressed? Concepts are a lot more nuanced than bare 1s and 0s because there's subjective meaning applied to them.
The complexity here is that concepts are generally recursive, they have intricate links to other concepts, which in turn link to others.
We could do the same thing with languages. Say a language is "English complete" if it can express at least everything expressible in English. I suspect all commonly used modern languages are English complete. The most obvious way I can see that a language would fail to be complete would be if it lacked nonce words or any equivalent device.
English is just one language, and while perhaps unusually expressive because of the amount of abuse it's received over the centuries, plus its ability to wholesale absorb new words on a whim, it is incapable of expressing some things with the same degree of succinctness and precision other languages have.
There are words in other languages that take a paragraph to explain in English because of all the cultural context they contain, or because they refer to things that most people who speak English have never seen, nor felt a need to describe with that degree of precision.
The problem with using English as a basis is it's very hard to explain in English what the language is incapable of expressing. This is like a fish trying to explain what they can do on land when the very concept of land is hazy at best.
This is how we can design a language for communicating general solutions to people's individual internal issues. Like self-help mad libs.
Plus we already have concepts we can't express. We can't share what looks like the resulting feeling in your mind of something the other never encounter, like a color to a blind man or a sound to a deaf girl. There are many feeling and sensations that you encounter in your life you can't define.
We can't define what we don't know either. E.G: we can't define whatever happen after death because we don't know about it.
At last, some practices, like meditation, make you (quite surprisely) discovers some very simple things, yet it seems quite impossible to share it using words with another person. Here knowledge and complexity are not the issue. It's just that labelling is not a silver bullet for communication.