“If anyone should ever succeed in deriving a real contradiction from Bayesian probability theory [...] then the whole edifice goes up in smoke. Along with set theory, ’cause I’m pretty sure ZF provides a model for probability theory.”
If you think of probability theory as a form of logic, as Jaynes advocates, then the laws and theorems of probability theory are the proof theory for this logic, and measure theory is the logic’s model theory, with measure-theoretic probability spaces (which can be defined entirely with ZF, as you suggest) being the models.
“If anyone should ever succeed in deriving a real contradiction from Bayesian probability theory [...] then the whole edifice goes up in smoke. Along with set theory, ’cause I’m pretty sure ZF provides a model for probability theory.”
If you think of probability theory as a form of logic, as Jaynes advocates, then the laws and theorems of probability theory are the proof theory for this logic, and measure theory is the logic’s model theory, with measure-theoretic probability spaces (which can be defined entirely with ZF, as you suggest) being the models.