You're going as far as Reinhardt cardinals? Maybe we can first get fine structural models for supercompact or extendible cardinals with choice?

Please don't cross post.

@Gro-Tsen: See (5) in my answer for that.

At the very least put a link to the other question for context...

@CalliopeRyan-Smith: I've added some stuff.

We use it because it's a powerful axiom that lets you prove stuff, and we know that it cannot add a new contradiction to the system, so we might as well take it with us. Some people are suspicious of it, since some people are suspicious of things they didn't build themselves. Some people are rejecting much more than the axiom of choice. This all had been discussed in very great lengths both on this website and on Mathematics in the past 14 years or so.

Well, at least for $n=1$ that's true. :-)

LS implies DC (and for countable languages it is equivalent), so it implies that every infinite set is Dedekind infinite!

Joel, sure, it's a nice lower bound. But as you say, you don't really know where that lower bound actually lies in the hierarchy of choiceless principles... That TC is strictly above ZF was already noted in the question!

Sure, but AC implies RR as well...

That's a nice question!