Comment by czgnome
10 days ago
In the article he says there is a model of ZFC in which the complex numbers have indistinguishable square roots of -1. Thus that model presumably does not allow for a rigid coordinate view of complex numbers.
10 days ago
In the article he says there is a model of ZFC in which the complex numbers have indistinguishable square roots of -1. Thus that model presumably does not allow for a rigid coordinate view of complex numbers.
It just means that there are two indistinguishable coordinate views a + bi and a - bi, and you can pick whichever you prefer.
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.
Haven’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.
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