In a discussion on a youtube video on the hydra game I jokingly mentioned how everyone was assuming that $\varepsilon_0$ was well-ordered. This lead to a bit of disagreement (in a nice way!) about the existence of the well-ordering of $\varepsilon_0$, assuming that $\varepsilon_0$ exists in whatever weak theory one was working in. The person with whom I was discussing seemed to think it obvious that if $\varepsilon_0$ existed, then it was an ordinal, hence well-ordered. I am very skeptical, to say the least. To clear things up, I said I would ask here:
Is it possible, in some suitably weak theory, to define (by a code, or outright) the object $\varepsilon_0$ whose elements (or 'elements') are the usual countable ordinals given by Cantor normal form using only smaller ordinals, but the well-ordering on $\varepsilon_0$ is not available in the theory? What is a good reference for the answer, in either case?
To me this seems like it should be true, but I can't pin down what the required weak theory is. Some form of second-order arithmetic seems reasonable to try, but I don't want one that implies the consistency of (first-order) PA, since then clearly $\varepsilon_0$ is well-ordered.