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.

▲

czgnome 2 hours ago | parent | next [-]

If 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.

	▲	yorwba an hour ago \| parent [-]
		They'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)).

▲

2 hours ago | parent | prev [-]

[deleted]