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:
States are vectors in Hilbert Space,
Evolutions are described by unitary operators and
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?