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Does Gödel's Second Incompleteness Theorem imply that no Theoretical Physics model of reality can be proved to be consistent using the laws of physics?

I work partially in Quantum Information Theory research which recently has had a resurgence of Quantum Foundations research themes. One of the themes of this field asks whether Quantum Mechanics, or more specifically, Quantum Information Theory is a fundamental theory of reality.

QIT is a particularly mathematical theory of reality, with highly mathematically phrased axioms:

  1. States are vectors in Hilbert Space,

  2. Evolutions are described by unitary operators and

  3. Von-Neumann's Projection postulate.

We can contrast these to other physical theory axioms, such as the rather physically based axioms of Relativity: equivalent reference frames and the frame independence of the speed of light.

Does Gödel's 2nd Incompleteness Theory imply that we cannot prove the consistency of our physical reality (at least as seen through the lens of QIT) using theory derived from QIT's axioms, or that if we do require consistency of QIT theory/experimental reality that we cannot ever prove that it's founding axioms are consistent?

How do we deal with this apparent paradox or anomaly? Surely it has implications for work into Quantum Foundations/the study of physics as a fundamental model of reality?

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  • $\begingroup$ You might get better answers in math.stackexchange.com. $\endgroup$ – Brionius Jan 6 '15 at 11:23
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    $\begingroup$ How so? Its implications did not seem obvious? $\endgroup$ – SLesslyTall Jan 6 '15 at 11:25
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    $\begingroup$ You shouldn't worry about Godel's Incompleteness Theorems. If QIT is a fundamental theory of reality then that means that Godel's theorem prevents us from proving some of its axioms, but the theory itself can be used to predict all experimental phenomena. In a vague analogy, you will have a language that can describe everything, but you will not be able to prove how every rule in the language arises using the language itself. $\endgroup$ – Constandinos Damalas Jan 6 '15 at 11:27
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    $\begingroup$ Axioms cannot be proved. The incompleteness theorem is about the fact that we cannot prove the consistency of the theory. But then, this is physically unimportant as long as the theory is not manifestly inconsistent and fits the experimental data. $\endgroup$ – Martin Jan 6 '15 at 11:31
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    $\begingroup$ Possible duplicates: physics.stackexchange.com/q/14939/2451 , physics.stackexchange.com/q/87239/2451 , and links therein. $\endgroup$ – Qmechanic Jan 6 '15 at 12:17