Maybe. Possibly at some point you cease being able to add non contradictory axioms that are also cannot be collapsed/simplified. ↩︎
Your original statement was correct. There are infinitely many non-isomorphic assemblies of axioms and inference rules.
For many systems (e.g. all that include some pretty simple starting rules), you even have a choice of infinitely many axioms or even axiom schemas to add, each of which results in a different non-contradicting system, and for which the same is equally true of all subsequent choices.
Commenting on the footnote:
Your original statement was correct. There are infinitely many non-isomorphic assemblies of axioms and inference rules.
For many systems (e.g. all that include some pretty simple starting rules), you even have a choice of infinitely many axioms or even axiom schemas to add, each of which results in a different non-contradicting system, and for which the same is equally true of all subsequent choices.