Gödel's incompleteness theorem prevents a universal axiomatic system for math. Is there any reason to believe that it also prevents a theory of everything for physics?
Edit:
I haven't before seen a formulation of Gödel that included time. The formulation I've seen is that any axiomatic systems capable of doing arithmetic can express statements that will be either 1) impossible to prove true or false or 2) possible to prove both true and false.
This leads to the question: Are theories of (nearly) everything, axiomatic systems capable of doing arithmetic? (Given they are able to describe a digital computer, I think it's safe to say they are.) If so, it follows that such a theory will be able to describe something that the theory will be either unable to analyse or will result in an ambiguous result. (Might this be what forces things like the Heisenberg uncertainty principle?)
Answer
If a "Theory of Everything" means a computational method of describing any situation, and true arithmetic formulas exist (as Gödel has shown) which cannot be proven, true arithmetic formulas exist which are necessary to describe some situation which cannot be discovered computationally, or if discovered incidentally, can not be proven true. So for example this computational method to be complete, would need to be able to prove the validity of math and logic, without using math and logic, since math and logic are separate from physics.
The definition above that "Gödel's theorem is a statement that it is impossible to predict the infinite time behavior of a computer program." is both incorrect, and anachronistic (at first Gödel rejected the Church-Turing definition of 'computability', but later (meaning by 1946) had to eventually discover it on his own). Besides Gödel wasn't a Computer Scientist even if his logic would be useful to them at some later date. The problem described above is a specific application of Gödel's theorem called the 'Halting Problem', but his theorem is much broader than that and it's implications much greater. What Gödel's first theorem basically states is that:
Any effectively generated axiomatic system $S$ cannot be both consistent and complete. In particular, for any effectively generated axiomatic system $S$ that is consistent which proves certain basic conclusions true, there are some basic true conclusion that is not provable within that system $S$.
For any formal effectively generated axiomatic system $S$, if $S$ includes a statement of its own consistency then S is inconsistent.
One of the answers above noted that:
- Gödel's theorem only applies to formal axiomatic systems (which is true)
However went on to suggest that "Almost no useful, real-world physical theories have ever been stated as formal axiomatic systems". This is completely false given how Gödel defined formal axiomatic systems. By formal axiomatic systems Gödel meant 'computable' meaning any system able to derive results (conclusions) through functions (or logic) that algorithmically computable. Physics completely relies on two such systems - Mathematics and Logic, which means Physics also is.
Is it really being suggested Physics is not computable? Physics makes predictions using math and logic, both of which are formal axiomatic systems. Physics also describes it's observed behaviour using the same systems. Physics is nothing less than a formal axiomatic system used to describe nature, though it does presuppose these other systems. Even if some of its axioms are observed or measured it derives results from these, or laws about them through functions that computable ($E=MC^2, F=MA$), therefore Gödel absolutely applies.
This means a Theory of Everything and indeed physics must either be internally consistent, but incomplete, meaning not actually able to describe every possible situation, or it must be complete but inconsistent, meaning able to describe every possible situation, but contain inconsistencies (self-contradictions). That physics requires mathematics to prove its own truths shows that physics is incomplete (since it needs to presuppose the consistency of Maths as an axiomatic system) just as mathematics requires logic to prove its theorems (for the same reason, Maths cannot prove logic, but must simply presuppose it). This is direct evidence of Gödel's claim that no axiomatic system can prove its own consistency, and so, is incomplete. Additionally, people have also shown that the Incompleteness theorem even holds in Quantum Mechanics (which is also consistent, but not complete).
Any 'Theory of Everything' cannot be complete since it cannot explain math, or logic, and there will be physical phenomenon whose behaviour cannot be computed. Just as physics itself, the physics of a TOE, in addition to physical observation, requires math and logic, showing how incomplete physics by itself is (though it is consistent).
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