Oh, by the way, it matters here that "weak" in the source you quote had a very specific technical meaning. If axioms are weak it means they apply to many things

Oh, by the way, it matters here that "weak" in the source you quote had a very specific technical meaning. If axioms are weak it means they apply to many things. Stronger axioms make more restrictions and apply to more specific things. So (arguably) axioms being weak is a good thing. Generally you want to work with axioms that are as weak as you can get away with.