Good point. I should have said "bizarre properties" or something to that effect. Which of course leads us to the problem of what exactly qualifies as such. (There's certainly lots of seemingly bizarre stuff easily derivable in ordinary real analysis.) So perhaps I should discard the second part of my above comment.

But I think my main point still stands: when there is no obvious way to see what choice of axioms (pun unintended) is "normal" using some heuristics like these, humans are also unable to agree what theory is the "normal" one. Differentiating between "normal" theories and "pathological" ones like PA + ~Con(PA) is ultimately a matter of some such heuristics, not some very deep insight. That's my two cents, in any case.