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> if all swans in the given universe ARE white, and you have no way of inspecting every one, you can never PROVE Yes, you can. Math is not physics. Pythagoras

> if all swans in the given universe ARE white, and you have no way of inspecting every one, you can never PROVE Yes, you can. Math is not physics. Pythagoras was able to prove his theorem about every triangle in the universe without inspecting every triangle in the universe. The trick is that "the universe" is not the physical universe, it's just a choice of axioms.

trashtester 4y ago
> Pythagoras was able to prove his theorem about every triangle in the universe without inspecting every triangle in the universe. My understanding is that statements about members of infinite sets can be decidable if there exists a method to set up use recoursion/induction to cover all of them. For the set of right triangles, this is relatively straight forward (if we ignore the complexity introduced by real numbers). For statements on infinite sets where it is not possible to reduce a proof to such recoursion/induction, you quickly end up needing an infinite number of steps to cover all cases, meaning the problem is undeciable/uncomputable. See the Church-Turing thesis: https://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis https://en.wikipedia.org/wiki/Church%E2%80%93Turing_thesis