Sorry if this is the wrong place, I've asked this question elsewhere, and I've been told this place is better for this.

Here is the idea: The universe obeys logical laws or it does not. If it does not, it's basically chaos, I guess, which is probably not very interesting here. So I'll discard that possibility. So, Let's assume the universe obeys logical laws.

Now, thanks to Gödel incompleteness, we know that any set of laws powerful enough to express Peano's arithmetics (basically natural numbers) will produce a set of undecidable propositions. And we won't be able to give an answer to these propositions within the initial set of laws we used.

Thanks to Leuven, who worked on Gödel incompleteness and randomization, we also know that the size of the set of undecidable depends on the complexity of the corresponding set of laws generating these undecidables. Basically, the more complex the theory, the relatively larger the set of undecidables. At the limit, when complexity tends to infinity, the probability that a proposition picked at random from a set of true proposition is undecidable tends to 1. Meaning, if you have an infinitely complex theory, then almost everything is undecidable.

Given that, if the universe obeys logical laws, it should also obey Gödel incompleteness. Does that mean the universe has many hidden truth that noone will ever be able to know ?

  • The universe might obey physical laws but not encode mathematical induction. So Gödel wouldn't apply. But a deeper issue is that if the universe "obeys" laws, where did those laws come from? That's the problem with Lawrence Krauss's "universe from nothing." Krauss defines "nothing" as quantum soup constrained by the laws of physics. Where do all those bubbling quarks come from, and where did the laws come from? Krauss is playing word games with the concept of nothing. – user4894 Feb 27 at 19:12
  • @user4894 this is just anthropic principle though. If the universe didn't allow for life we would't be here to ask why it seems so perfect for life. There may be many possible initial stayes of the universe that allow for life.. and when we know how to simulate universe creation, perhaps it'll turn out that the universe isn't as fine tuned as it appears to be. – Richard Feb 27 at 19:51
  • "if the universe obeys logical laws, it should also obey Gödel incompleteness" Why? – Eliran Feb 27 at 21:07
  • @Eliran This requires the Church/Turing thesis, but many people accept that. The reason is because if the universe obeys logical laws, deciding what happens next is exactly equivalent to a complex computation. If all forms of computation are Turing Machines, then nature obeys First Order Logic, which can model Turing Machines. – jobermark Feb 27 at 21:14
  • Not obeying deterministic laws does not imply chaos, as long as you also have the Law of Large numbers. Quantum indeterminacy does not destroy the applicability of all physics. It just means certain kinds of measurement are impossible. Perhaps all the unprovable statements, true and false, require accuracy that is impossible according to Heisenberg's Principle in order to be determined. The we already know we can't decide them, and we don't have to try. – jobermark Feb 27 at 21:17

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