Theorem. If ZFC is consistent, then there is a model of ZFC that has a definable complete ordered field ℝ with a definable algebraic closure ℂ, such that the two square roots of −1 in ℂ are set-theoretically indiscernible, even with ordinal parameters.
replyHaven’t thought it through so I’m quite possibly wrong but it seems to me this implies that in such a situation you can’t have a coordinate view. How can you have two indistinguishable views of something while being able to pick one view?
Mathematicians pick an arbitrary complex number by writing "Let c ∈ ℂ." There are an infinite number of possibilities, but it doesn't matter. They pick the imaginary unit by writing "Let i ∈ ℂ such that i² = −1." There are two possibilities, but it doesn't matter.
replyIf two things are set theoretically indistinguishable then one can’t say “pick one and call it i and the other one -i”. The two sets are the same according to the background set theory.
replyThey're not the same. i ≠
−i, no matter which square root of negative one i is. They're merely indiscernible in the sense that if φ(i) is a formula where i is the only free variable, ∀i ∈ ℂ. i² = −1 ⇒ (φ(i) ⇔ φ(−i)) is a true formula. But if you add another free variable j, φ(i, j) can be true while φ(−i, j) is false, i.e. it's not the case that ∀j ∈ ℂ. ∀i ∈ ℂ. i² = −1 ⇒ (φ(i, j) ⇔ φ(−i, j)).
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