However, I claim that these paradoxical sentences don't present any real problem for our capacity to understand reality, or even to represent it in a formal system. I have to get a bit technical to demonstrate what I mean, sadly, so bear with me. Suppose that we have a giant table of 1s and 0s, representing the full history of reality, the full history of its physical configurations. Each column of the table represents the world as it stands at a given instant in time; each zero represents answering "no" to some question about the state of the world, and each one represents answering "yes" to some question about the state of the world. In this framework, modulo concerns about reality having infinitely many details, you should be able to exhaustively describe every state the universe will ever be in.^[3]

Now, if we introduce logic gates into this framework, we can also use it to represent the laws of physics, or the rules for evolving the state of reality over time. (This isn't unlike applying rules of inference in formal logic, an observation we'll return to later.) You can imagine a massive series of logic gates connecting each column in our table to the one immediately to its right; the gates could be set up such that, by simply inputting one complete configuration of reality at the start, later configurations could be derived automatically by a cascade of mechanically triggered logic gates.^[4]

Between the "bits as answers to yes-or-no questions" frame and the capacity of logic gates to evolve the values of bitstrings in arbitrary ways, I think it's clear that this formal system can in principle be used to represent the physical world exhaustively, or at least that you'd only fail due to not having access to enough physical hardware (or knowledge of the universe) to represent reality using this method. So that seems to blow a hole in the idea that the liar's paradox and/or Gödel's theorem prove that we're incapable of understanding the material universe, or even building formal systems which represent it.

Is the point of this whole thing that if you have a finite system that evolves according to a finite set of rules for a finite number of steps, then you can prove anything about the system?