Earlier quoted context omitted.
Aren't you confusing language with understanding? In computer terms (which I'm more familiar with) this is the difference between expressing a program in some language vs evaluating that program.
A Turing machine is a very simple device. Evaluating if something is "Turing complete" is simply determining if it can function like a Turing machine. 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…
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.