[danharaj]

Even computable reals do not have computable ordering.

> but presumably they lie somewhere regardless of my inability to do it on a TM.

Why?

> Zermelo's theorem at that point right?

It is declared by fiat in standard set theory that infinite sets can be well ordered. This is no real mathematical justification. The real justification is social: that it is convenient for mathematicians to not care about the ontology of these nasty infinite objects so long as results are mostly reasonable for objects that mathematicians actually care about. You don't get into too much trouble pretending the reals are nice so long as you don't look too hard.

[rrmm]

I mean that's Wildberger's whole point isn't it?

[danharaj]

Yes, I'm just trying to emphasize that there is real serious mathematics behind his point, it's not just a matter of philosophical taste.