I'm not asking about teaching or textbook-writing; my question is a tangent to that discussion. Smoofra invoked a notion of "impact"; I'm trying to determine how smoofra rates Cox's theorem on that scale.

I know of a number of paths to the Bayesian approach:

• Dutch book arguments (coherence of bets)
• more elaborate decision theory arguments in the same vein (coherence of decisions under uncertainty)
• the complete class theorem (the set of decision rules "admissible" (non-dominated) in a frequentist sense is precisely the set of Bayes decision rules)
• de Finetti's theorem (exchangeability of observables implies the existence of a prior and posterior for a parameter as a mathematical fact)
• Cox's theorem (a representation of plausibility as a single real number consistent with Boolean logic must be isomorphic to probability)

My question is about which justifications smoofra knows (in particular, have I missed any?) and what impact each of them has.

FWIW, in high school my very first introduction to integrals used the high-school version of Riemann integration with a simple definite integral that was solved with algebra and the notion of the limit of a sequence. Once it was demonstrated that the area under the curve was the anti-derivative in that case, we got the statement of the Fundamental Theorem of Calculus and drilling in anti-derivatives and integration by parts etc.