Edit: upon reflection, I think maybe you and I are interpreting "The set of natural laws" in the GGP differently. I think maybe maybe you're interpreting this as a statement about how the universe actually works, instead of a statement about the set of laws enumerated by modern science (or any logic-based successor of modern science). The GGP mentioning the axioms of the laws however, makes it clear they're talking about the set of laws discovered and discoverable via the scientific method.
In other words, there are two ways to talk about the laws of nature: (1) some set of Platonic ideal laws existing outside of human experience, that actually govern the universe and (2) the set of approximations of these ideal laws that could possibly be discovered by a logic-based science. The GGP's mentioning of axioms means they're clearly talking about (2), but your statement makes much more sense if meant about (1).
Everything below is based on my original understanding of what you wrote, which I believe is a misunderstanding on my part.
I'm not sure what you're getting at with regard to Godel's incompleteness theorem not applying to modern science.
Are you arguing that Godel's incompleteness theorem doesn't apply to the mathematical logic model(s) at the heart of the modern scientific method, because a scientific model is an approximation of reality? The match between the models and reality has no bearing on the limits of the structure of the models themselves.
The original statement was about the limits of the logical structure of the models underlying modern science, having nothing to do with how well they actually fit reality.
Are you perhaps saying something along the lines that because scientific models don't precisely match reality, it's okay to introduce axioms that make them complete (able to prove all true statements in their domain) but inconsistent (and therefore able to construct proofs that false statements are true)?
I dispute that knowledge obtained through scientific enquiry is rigorously based on mathematics, and therefore I do not believe that results about mathematical provability necessarily imply anything about the limits of scientific knowledge.
Gödel's work was concerned with formally defined abstract systems, and his results demonstrate the limits of mathematical proof within such systems.
But science never proves anything. Science often uses the language of mathematics to express and to quantify ideas, but the core of science is observation, hypothesis-forming, and experimentation. Scientists apply logic to rule out theories, but it's an informal application of logic, not a formal one, because you can never precisely define a theory the way you can precisely define a mathematical object.
Science may appear to be a rigorous discipline, but it is at best diligent, not rigorous - not in the sense that a proof of a mathematical theorem is rigorous. A proven theorem must necessarily be true - that's what the proof demonstrates. Meanwhile, a scientific theory is only conditionally true, inferred from the evidence and hypotheses about the underlying mechanisms of the universe. All scientific truth is subject to revision if contradictory evidence arises.
Science is a system of best guesses based on what we have observed. Some of those guesses have proven very useful, and very reliable at predicting the future. But none of those guesses are fundamentally based on, or necessarily limited by, formal axiomatic systems of logic.
For all we know, there is a low upper bound on the complexity of the universe, and it might be completely explainable and understandable through the lens of science, without getting anywhere near the towering near-infinities of abstract mathematical thought.
Alternatively, if the universe was a formal system about which there were unprovable truths, one could simply add that unprovable truth as an axiom, to form a larger formal system, which itself would have unprovable truths, but as long as the universe was entirely contained within that larger system, you could prove every truth relevant to it, without being limited by Gödel.
If someone is seeking unknowables in the real world, something like the uncertainty principle could be a closer match.
In other words, there are two ways to talk about the laws of nature: (1) some set of Platonic ideal laws existing outside of human experience, that actually govern the universe and (2) the set of approximations of these ideal laws that could possibly be discovered by a logic-based science. The GGP's mentioning of axioms means they're clearly talking about (2), but your statement makes much more sense if meant about (1).
Everything below is based on my original understanding of what you wrote, which I believe is a misunderstanding on my part.
I'm not sure what you're getting at with regard to Godel's incompleteness theorem not applying to modern science.
Are you arguing that Godel's incompleteness theorem doesn't apply to the mathematical logic model(s) at the heart of the modern scientific method, because a scientific model is an approximation of reality? The match between the models and reality has no bearing on the limits of the structure of the models themselves.
The original statement was about the limits of the logical structure of the models underlying modern science, having nothing to do with how well they actually fit reality.
Are you perhaps saying something along the lines that because scientific models don't precisely match reality, it's okay to introduce axioms that make them complete (able to prove all true statements in their domain) but inconsistent (and therefore able to construct proofs that false statements are true)?