Hmmm...

::goes and learns some more math from Wikipedia::

Okay... I now know what an ordinal number actually is. And I'm trying to make more sense out of your comment...

So, re-reading this:

or even be uniquely specified by second-order axioms that pin down a single model up to isomorphism the way that second-order axioms can pin down integerness and realness, is something we have rather less evidence for

So if I understand you correctly, you don't trust anything that can't be defined up to isomorphism in second-order logic, and "the set of all countable ordinals" is one of those things?

(I never learned second order logic in college...)