[drdeca]

If you want to add axioms in order to be able to show each of an infinite number of statements which are all independent of the initial axiom system, and also none of them follow from the rest of them, that could involve an infinite set of axioms, yes,

but any statement we could make about stuff in physics, if it could be expressed in the language of the formal system, would, as a single statement, be something that could be added as a single axiom.