Lambda the Ultimate

I think that it can safely be said that the natural languages can transport any formal structure; that we can communicate any mathematical structure using the natural language. That's the essence of metamathematics.

But then we are led to believe that the natural language has no proper formal structure. It is informal. In the sense that it is not possible to feed the Don Quijote to an algorithm that will be able to pinpoint any formal inconsistency implicit in the text (without recurring to any other text or knowledge). An example imaginary inconsistency would be if at one point Cervantes says that Don Quijote always likes lo love Dulcinea, and at another he says that he does not like to love her. The kind of inconsistencies that would destroy any formal structure if inadvertently injected in transit.

These 2 previous paragraphs seem a bit paradoxical to me, since, if there is no procedure to decide whether a natural text is inconsistent (or if there are only heuristic fallible procedures) metamathematics should be impossible, and mathematics could not have been born.

If we take a formal structure expressed in some mathematical formalism, and express it with the natural language, metamathematically, we need to be sure that there are no inconsistencies in any of the cases. In the former case, we can check, and there are algrithms that can check. In the later case, we can check. Can no algorithm generally check?

I know that there is a history to this, from Llull and Leibniz, reaching a summit with Frege, to be toppled by Russell, and followed by the neopositivists etc., and later experiments like the semantic web.

So my question here is: Is there some kind of proof or argument showing that the natural languages cannot be provided with a simple mathematical foundation? Some recognizable fundamental property of the natural language that is inherently inconsistent? (Are both questions the same?)

What I mean with a "simple" mathematical foundation is that it must consist on a core formal theory (that can then be suplemented with a number of ad-hoc rules that provide for shortcuts and phrases in the natural languages). A priori, it should be totally disprovided of semantics. Then we should be able to interpret extensions built on that foundation in natural language texts, taken as abstract structures of words; and then, by proxy, we might provide those extensions with the semantics of the natural language texts in which they are interpreted. (Perhaps we need some a priori semantics, for things like time [edited to add: or I? That would seem to lead us towards Gödel's pit, damm. However, I think that we can settle in principle for a descriptive language without person], that are deeply involved in the basic structure of the natural language?)

So, simple in the sense that Frege's proposal was simple, or that the semantic web is simple (my previous paragraph is meant to be interpreted in their intended use), but a huge neural network with crazy amounts of delicately balanced branches trained by all books ever published is not simple. Notwithstanding the problem that with the neural network solution we are dealing with an ungodly mixture of syntax and semantics.

I really don't know whether I am using the right terminology to phrase my question - or even whether there exists a terminological framework where it can be meaningful and exact. So apologies if I have taken a couple of poetic licences in trying to lay it out.