One argument that is often raised against hypothetical frequentism, as e.g. developed by Reichenbach or Von Mises is that limiting relative frequencies can violate the axiom of countable additivity (see e.g. Alan Hajek 15 arguments against hypothetical frequentism) and are therefore not admissible. Where admissible is a requirement brought forward by Salmon, which demands that: "We say that an interpretation of a formal system is admissible if the meanings assigned to the primitive terms in the interpretation transform the formal axioms, and consequently all the theorems, into true statements. A fundamental requirement for probability concepts is to satisfy the mathematical relations specified by the calculus of probability… "
I would question however that this is really a valid requirement. If the interpretation of probability as limiting relative frequency is helpful and well-working in many cases, why should these be spoiled by the fact that it fails in some, possibly irrelevant cases? The following example might clarify my idea: If certain assumptions are met, I can interprete linear regression as causal relationships between variables. For this to be a valid interpretation it is however irrelevant whether there are causal relationships that do not fit into the framework of linear regression. The same holds for hypothetical frequntism: Just because there are imaginable scenarios with arising limiting relative frequencies that do not fit into Kolmogorovs axiomatization it does not mean that I can validly interprete probability as limiting frequencies in many other cases. Do you think I have a valid point or am I missing an important argument for the admissibility criterion?