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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 ?

davcha
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  • 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 '19 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 '19 at 19:51
  • "if the universe obeys logical laws, it should also obey Gödel incompleteness" Why? – E... Feb 27 '19 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. –  Feb 27 '19 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. –  Feb 27 '19 at 21:17
  • Logical laws apply to collections of propositions or statements, not to the universe. So the universe can not obey logical laws any more than it can obey the English grammar. That a proposition is undecidable does not mean that we can not find out if it is true or not, it just means that it or its negation can not be derived from a particular set of axioms. The undecidable Gödel sentence from his theorem is, in fact, known to be true. So no, Gödel incompleteness tells us nothing about the universe. – Conifold Feb 27 '19 at 21:33
  • @Conifold But those logical 'laws' would presumably serve as axioms. I guess you need to strengthen this by clarifying there are finitely many of them statable in in finite form, etc; giving a definition of 'obeys'; and adding the Church/Turing thesis. But it is not as silly as it seems. Without uncertainty, the watchmaker notion of physics, where the universe really obeys and embodies the predictions of some finite theory, which predicts it perfectly, would, I think, actually be paradoxical. –  Feb 27 '19 at 21:51
  • Possible duplicate of What is the connection between conscious mind and Gödel's incompleteness in a mathematical universe? – Conifold Feb 27 '19 at 21:58
  • @jobermark regarding deterministic laws. I haven't said a thing about determinism. In fact, Gödel incompleteness remain valid even if you talk about probabilities and stochastic processes. That being said, I kind of like that idea about uncertainty principle, but I'm not sure I really understand how that uncertainty would fix the paradox. – davcha Feb 28 '19 at 01:03
  • Natural laws are deterministic by nature. How do you 'obey' something that does not determine your behavior? 2) Consider how Goedel's proof goes. If all your numbers have uncertainties, when you go about computing Goedel numbers, the uncertainties will be immense, and you will not be able to tell one formula's coding from another's. Basically, you need the ability to be very certain all your mathematical operations remain integers in order for arithmetic to give you trouble. It is the discrete nature of logic that makes it incomplete.
    • –  Feb 28 '19 at 05:32
  • "logical laws" ? The laws of logic are about... logic, i.e. language, i.e. human thinking and activities. Thus, it makes little sense to say that "the universe obeys them". – Mauro ALLEGRANZA Feb 28 '19 at 07:14
  • Godel's Theorems are about formalized systems of arithmetic, not about physical universe. – Mauro ALLEGRANZA Feb 28 '19 at 07:14