Hilbert's Sixth Problem: Can Physics Be Completely Axiomatized?

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In summary, the conversation discusses the potential for a complete axiomatization of physics and whether Godel's incompleteness theorem would apply to such a system. The conversation also touches on the implications this would have for Hilbert's sixth problem and the "theory of everything." It is noted that the current understanding is that Godel's theorem would only apply once all the axioms are established, and it would mean that there are some things that are undecidable within the system. Further discussion and resources on this topic are also provided.
  • #1
( 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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  • #2
What is your interest in Hilberts sixth problem? Are you working on a homework paper? Or is this just basic curiosity?

For starters


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.


Here's some more discussion about the two


What is Hilbert's sixth problem?

Hilbert's sixth problem is one of the 23 unsolved problems in mathematics proposed by German mathematician David Hilbert in 1900. It focuses on finding a mathematical proof for the existence of solutions to a system of algebraic equations with a finite number of variables and equations.

Why is Hilbert's sixth problem important?

Hilbert's sixth problem is important because it addresses fundamental questions about the nature of mathematical truth and the limits of mathematical knowledge. It also has connections to other areas of mathematics, such as algebraic geometry and topology.

Has Hilbert's sixth problem been solved?

No, Hilbert's sixth problem has not been fully solved. However, significant progress has been made in the field of algebraic geometry, and many special cases of the problem have been solved.

What are some approaches to solving Hilbert's sixth problem?

Some approaches to solving Hilbert's sixth problem include using techniques from algebraic geometry, topology, and logic. Other approaches involve developing new mathematical theories and tools to tackle the problem.

What impact would solving Hilbert's sixth problem have?

Solving Hilbert's sixth problem would have a significant impact on mathematics, as it would provide a deeper understanding of the nature of mathematical truth and the structure of algebraic equations. It could also have practical applications in fields such as computer science and physics.

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