While wandering about the ability of Large Language Models to understand, and the Chinese room argument (related question Is the "Chinese room" an explanation of how ChatGPT works?) I got into a mess with the very idea of understanding. Understanding something external to understanding looks like possible: we can somehow understand an ant by observing it, its shape, its behaviour... We study an object with a different object with no overlap.
Now, how about attempting to understand everything? In particular, what happens when one wants to understand understanding?
A reasonable object for understanding are understandable things. In particular, if understanding is understandable, a greedy agent would try to understand understanding. Now, if understanding is always understandable, understanding understanding is also understandable, so now the agent would be after understanding understanding understanding. Since for any order of understanding one can define a higher order understanding, there is no possible end to understanding, even with the single object of study $understanding$.
I am not sure whether we should compare this infinite chain of embedded $understanding^k$ to the natural numbers, where there is a final support (zero) for anything to be understood; or to the integers, where there is no first integer number to the left and thus no possible end to fully understanding understanding.
Is complete understanding impossible?