Comment by skissane

> "Cantor's diagonalization argument" is best understood as a mere special case of Lawvere's fixed-point theorem. Lawvere's theorem is really the meat of the argument, and it's also the part that is very easy for exotic systems to "accept", since it's close to a purely logical argument.

Okay, but can you prove Lawvere’s theorem in NFU+AxCount?

And even if you can, since NFU+AxCount proves that the reals are countable, if NFU+AxCount proves Lawvere, then (to echo what btilly says in a sibling comment) NFU+AxCount+Lawvere couldn’t entail the countability of the reals, since that would render NFU+AxCount trivially inconsistent, and we know it is isn’t trivially inconsistent (as with any formal system, consistency is ultimately unprovable, but if a system is taken seriously as an object of mathematical research, then any inconsistency must be highly non-trivial.)