How isormorphic it is remains to be seen. The infinite set digressions have not been particularly helpful to real problems.

The objective bayesian is free to estimate frequencies, and has done so, a la Jaynes. He explicitly identifies that both questions are answering different questions, and answers both.

I'm not aware of anyone doing this, but I think a frequentist could just as well interpret subjective degrees of belief in frequentist terms, but the sample space would be in informational terms, looking for transformation groups in states of knowledge.

"Probability" is a word used in the interpretation of probability theory.

Sometimes. I think if we're trying to keep terms straight, you should separate probability_SubjectiveBayes, probability_Math, probability_Frequentist, and probability_HumanLanguage. You seem to conflate probability_Math and probability_HumanLanguage.