Yes, that would fit in what I am talking about. I have a bad habit of constantly editing posts as I write, so you might have seen my post before I wrote this part.

I don't necessarily think we can find such a set of axioms. mind you. I can't guarantee that there are a finite number of statements required, or that the human mind is necessarily is capable of producing/comprehending said statements, or even that any mind stuck within the constraints of the universe itself is capable. (I suppose you can take issue with the use of the word "describable" at this point). But I do think the statements exist, in some platonic sense, and that if we buy into logic we can at least know that they exist even if we can't know them directly. (In the same sense that we can often know whether or not a solution exists even if it's impossible to find)

Such a universe wouldn't even necessarily be "complicated". A single infinite random binary string requires an infinitely long statement to fully describe (but we can at least partially pin it down by finitely describing a multiverse of random binary strings)