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by Xcelerate
722 days ago
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> Yet, I know no consequences that would be measurable in physics One can build a physical device modeled off of a Turing machine that enumerates all proofs within ZFC. The machine halts if an inconsistency is discovered, and runs forever if not. Now a prediction can be made about a process in the physical universe whose outcome depends on the axiom of choice. I’m not trying to sound facetious actually. Highly abstract mathematics plays a critical role in inductive inference (in the sense of speeding up universal search by mapping a search over program space to a search over proofs in formal systems). This appears to be the direction some recent ML research is heading, so it wouldn’t surprise me if a lot of “unphysical” axioms end influencing our ability to efficiently approximate Solomonoff induction. |
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This is because we can prove, in the small and generally trusted metatheory PRA, that ZFC is inconsistent if and only if ZF (= ZFC − AC) is inconsistent (if and only if IZF (= ZFC − AC − LEM) is inconsistent).
[ This metaproof rests on the fact that ZF can prove that the axiom of choice (AC) holds in "Gödel's sandbox" L, the "constructible universe", even if it might not hold in the universe of all sets. ]
In other words: Adding the axiom of choice to ZF doesn't cause new inconsistencies. In case ZF is consistent (a statement which most logicians believe), then ZFC is so as well.
A couple pointers to the literature are here: https://www.speicherleck.de/iblech/stuff/37c3-axiom-of-choic...