A Proof that there's No Constructive Proof of the Intermediate Value Theorem

The other day my friend Lucas Salim was asking me some questions about categorical logic and constructive math, and he mentioned he’d never seen a proof that there’s no constructive proof of the intermediate value theorem before. I showed him the usual counterexample, and since my recent blog post about choice was so quick to write I decided to quickly write up a post about this too, since I…