No, commonly the key to "some claims about the real world" is independence. Then can apply, with meager assumptions, say, the weak law of large numbers (which

No, commonly the key to "some claims about the real world" is independence. Then can apply, with meager assumptions, say, the weak law of large numbers (which we do prove), that is, take an average. Also sometimes can make some "claims' based on conditional independence (which my axioms give us the ability to define) and apply, say, the martingale convergence theorem.