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by mauricioc
2537 days ago
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One exciting new development is "Cubical Type Theory: a constructive interpretation of the univalence axiom" [0], a constructive version of homotopy type theory. The "axiom of choice" in this context is not really an axiom, just a provable theorem whose proof in this setting is given by Martin-Löf. The reason, I believe, is that constructive mathematics has a different meaning for the exists (∃) quantifier. In a constructive context, existence is "stronger" since you have to specify a witness of existence; this makes the hypothesis of the "axiom" of choice stronger, allowing for a proof. Here's a stab at a very informal version of this. Suppose you have a set composed of "slices". The "axiom" of choice says that "IF for every slice (x in A) there exists a good element of that slice (y in B_x, and 'good' is defined by satisfying property C), THEN there exists a function which takes a slice and returns a good object of that slice." But the very definition of the (constructive) word "exists" in the hypothesis means you already have to specify, as input, a way of finding one good object of each slice. With this, finding a function that provides a good object of each slice is a tautology. [0] https://arxiv.org/abs/1611.02108 |
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