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

  • Thread starter Austen Bentley
  • Start date
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
Austen Bentley
1
0
( 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 ).

Discuss
 
Last edited:
Physics news on Phys.org
  • #2
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/qu...h-problem-the-axiomatization-of-physics-after
 

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

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.

Similar threads

On soundness and completeness of ZFC set theory
    • Set Theory, Logic, Probability, Statistics
Replies
6
Views
2K
A The Blessings of Dimensionality and Hilbert's 6th Problem
    • Set Theory, Logic, Probability, Statistics
Replies
4
Views
1K
I An easy proof of Gödel's first incompleteness theorem?
    • Set Theory, Logic, Probability, Statistics
Replies
16
Views
2K
MHB Gödel Number Usefulness to Determine an Infinite Set's Completeness
    • Set Theory, Logic, Probability, Statistics
Replies
1
Views
1K
A Formal axiom systems and the finite/infinite sets
    • Set Theory, Logic, Probability, Statistics
Replies
10
Views
2K
A confusion about Godel theorem and real numbers
    • Set Theory, Logic, Probability, Statistics
Replies
7
Views
2K
A A "Proof Formula" for all maths or formal logic?
    • Set Theory, Logic, Probability, Statistics
Replies
3
Views
2K
I Can interpretation-dependent facts be derived from the axioms?
    • Quantum Interpretations and Foundations
Replies
10
Views
2K
The Current State of Mathematical Rigour
    • General Math
Replies
6
Views
1K
Is Alan Turing given too much credit when it comes to computers?
    • Programming and Computer Science
Replies
29
Views
3K

Hot Threads

Recent Insights

Back
Top