Is there an agreed upon convention in general for what to name ZFC+[Large Cardinal Axiom]? Or would one have to state explicitly which axiom was being added?
To explain what I mean, note that anyone who has taken a course in axiomatic set theory will know which axiom is being added when going from ZF to ZFC.
If, for example, one adds the axiom (roughly stated) "for every cardinal there exists a strictly larger inaccessible cardinal" then is there an abbreviation (perhaps ZFC+I?) that any set theorist would recognise? If not for my example, would such abbreviations exist for more notable or more commonly seen axioms?
I think that whenever I saw anyone using anything stronger than $\sf ZFC$, the axioms were defined explicitly. Except, perhaps, things like $I0,I1,I2,I3$ which are essentially the conventional names for these statements.
Most of the time you write either something like "We abbreviate by $\sf IC$ the statement "There exists an inaccessible cardinal", and in the following section we will work with the theory $\sf ZFC+IC$"; or you just write the axiom explicitly each time along the lines of "Assume that $\kappa$ is a measurable limit of Woodin cardinals, then ..."