Gödel’s theorem may be demonstrated using arguments having an information-theoretic flavor. In such an approach it is possible to argue that if a theorem contains more information than a given set of axioms, then it is impossible for the theorem to be derived from the axioms. In contrast with the traditional proof based on the paradox of the liar, this new viewpoint suggests that the incompleteness phenomenon discovered by Gödel is natural and widespread rather than pathological and unusual.

He proves Godel’s Theorem using Turing’s Halting Problem.

The other is a powerpoint presentation by Avi Widgerson at the Institute for Advanced Study in Princeton, titled “Proof, Randomness, Computation, Games” that provides a nice high level overview.

To me as an empiricist, this is mysticism. We know nothing about “the Universe”, except what we know from observation, and the theories that we construct and have worked so far.

What that wider reality consists of, we gradually discover by exploration. That is not a Godelian problem it is a Bayesian one. Godel is only telling us that no one theory we invent will be able to solve every problem. It can be the simplest little theory (almost) or subtheory, or the most complex.

Unless a theory is so simple that it is Godel-complete, then somewhere within the theory there are true statements that cannot be proved from within the theory. i.e. We will never reach the point when we don’t need new theories - unless we find a ToE which is so simple that it is indeed G-complete, and from that (finally) we will be able to find (hack) emergent theories that can explain everything.

I can’t emphasise this strongly enough: Physics is a creation myth. A very good creation myth - one that agrees with all the known evidence. Some of us find it a very exciting creation myth. But it is first and foremost a narrative we are building as we go. And we, as storytelling humans know nothing, except what we observe, and the myths (models) we construct to explain what we see.

we have to be careful not to confuse “statements of truth” with “computation”. Godel addresses the provability of statements. Statements are things like “There are infinitely many primes” or “even numbers are the sum of two primes”. These may or may not be true. The provability of these statements, according to Godel, is the thing we cannot take for granted. (Euclid proved the first above. The second is Goldbach’s conjecture, for which I am not sure if a proof has yet been found.)

A computation on the other hand is a process whereby inputs are fed into a function, and the result are outputs. The function must be finite, can not be magical, and can only perform basic operations. The definition of these operations relies heavily on the “space” in which the function operates - in the example of Monte Carlo simulations, which incidentally is just a fancy term for “randomize the inputs”, the basic operations are simple arithmetic. I have used Monte Carlo extensively in my own research and training and its nothing magical.

However! not every function is necessarily computable. In fact that statement is called Turing’s undecidability Theorem. It is tempting here to simply say “Godel = Turing!” and that functions equal statements but these concepts are not homomorphic, let alone isomorphic.

Now, you could define the n-body problem as a series of statements (including Theorem Duck: Planet X is at Position Rho at Time Tau). Let us call this collection of statements the N Body Problem System. If you do this, then yes, the solution to the N body problem will be outside that set of statements describing it. And yes that is due to godel - but that is because we have defined a very specialized subset of the Universe, focused soely on the N body problem. So of course Theorem Duck’s provability *within the N Body problem System* is not guaranteed. However that doesnt have any bearing on whether the N Body problem itself, in general, is godelian in hte much larger system called the Universe, which contains the N Body problem as a subset, and which Theorem Duck might well be contained within, and thus provable. Or maybe not.

The reason we can solve the N Body problem in the real universe using Monte Carlo is because classical mechanics can be described using Turing Machines, such as the MATLAB programming language. Or slide rules. Both qualify. Practically every single physical theory meets the conditions for computation (the “Church-Turing Thesis”) and so we can do numerical simulations to model them (though our simulations are never as perfect as the Real Thing, because to do a perfect simulation, we would need a simuation of the same complexity as the actual system, namely a pocket Universe like you might find in Zarniwoop’s office. So we can’t really “solve” the N Body problem anyway, but we can get as arbitrarily close to figuring out what we need to know for pragmatism’s sake.

I am so not a theoretical physicist, so everything i said above might as well be Godelian. It certainly may be true.

There are a lot of problems like that in fluid dynamics. We can run simulations, with controlled random components. We can rerun them hundreds of times and statistically examine the results and determine what reality probably will look like. (Assuming the simulation isn’t flawed, a really big assumption.) But we cannot derive formulas which will directly predict the result. So we know what the answer is, and we know that the answer is right, but we can’t prove it within the system.

A godelian example might be that we observe the orbit of a planet and find that taking everything into account the planet should be at position X,Y,Z but instead appears to be offset by dX,dY,dZ. That actually happened, but then we used relativity to reduce the delta to zero. ut suppose we do a more fine grained measurement and find again some offset that cannot be explained. All godel says is that there is no guarantee that we will find another theory to fill in the gap and again reduce the deltas to zero. Personally i would be predisposed to believe that a theory does exist simply because things are always easier at the macroscopic level. Its when you drill down to microscopic scale and below that theres more potential for running into fundamental limits. This is why string/M theory may be closer to some godelian truth than mere celestial mechanics. The latter are really just the integral of the former, in a sens e- an aggregate behavior, like a mob.

We have no reason to believe they don’t work at this scale and speed.

In fact, the fact that you can find approximation schemes that are able to solve the physical problem to arbitrary accuracy, numerically on a computer say, proves that the underlying laws of physics are OK.

Now it is the laws of physics that are the axioms that are used to create the dynamic equation of the Three Body Problem. It’s just that the equations have no analytical solution, at least not internally from within the theory.

You have to jump out of the system and use an approximation scheme to get around this absolutely intractible roadblock.

If anyone knows why this is wrong please forgive. Like I said I am very rusty with this stuff.

But in the Three Body Problem, we don’t actually know the answer, and have no way of figuring it out.

Godel predicts that there are propositions that are true but can’t be solved.

The TBP is clearly Newtonian gravity- there’s nothing else involved. But the math is unsolvable.