Hilbert's sixth problem

  • #1

Main Question or Discussion Point

( note ⊕ is the mutually exclusive or ) First note trivially an axiomatiziation of physics would be complete as [itex] (\forall \phi( \phi \in L)) [/itex] s.t.L ↔(the language of the formal axiomatization of physics) the statement ϕwould be an arbitrary statement of some physical property or event etc and thus by the definition of the axiom system one should be able to prove [itex] \phi \oplus \neg \phi [/itex] .

If one assumes it is possible to axiomatize physics completely (a correct solution to Hilbert's sixth problem) could one then use Godels incompleteness theorem to prove that there are some parts of the physical universe that we can never truly understand(prove from this purportedly complete axiom system) thus contradicting the statement that the axiomatization is complete thus proving (informally) a negative solution to hilberts sixth problem? (One assumes the complexity of the axiom system is strong enough to prove basic arithmetic properties as required by the incompleteness theorem.)

Wouldn't this also prove the "thery of everything impossible as well, as it could be treated as an axiom system (even if it is a single equation, as it could be a single axiom) and algebraic manipulations would be derivations from (theorems of) the axiom system and thus by godels incompleteness theorem would also be incomplete and thus not truly a "theory of everything" as there exist physical phenomena that could not be proven or disproven within the system(or both and thus is inconsistent ).

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Answers and Replies

  • #2
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What is your interest in Hilberts sixth problem? Are you working on a homework paper? Or is this just basic curiosity?

For starters

http://en.m.wikipedia.org/wiki/Hilbert's_sixth_problem

From the wiki article there's no mention of completeness only the axiomization of physics. So it seems that Godels theorem would only apply once we had all the axioms and then it would mean that there exists some things which are just undecidable in the system using those axioms.

http://en.m.wikipedia.org/wiki/Gödel's_incompleteness_theorems

Here's some more discussion about the two

http://physics.stackexchange.com/questions/87239/what-happened-with-hilberts-sixth-problem-the-axiomatization-of-physics-after
 

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