You might be interested in some work by Glenn Shafer and Vladimir Vovk about replacing measure theory with a game-theoretic approach. They have a website here, and I wrote a lay review of their first book on the subject here.
I have also just now discovered that a new book is due out in May, which presumably captures the last 18 years or so of research on the subject.
This isn’t really a direct response to your post, except insofar as I feel broadly the same way about the Kolmogorov axioms as you do about interpreting their application to phenomena, and this is another way of getting at the same intuitions.
I clicked this because it seemed interesting, but reading the Q&A:
In atypical game we consider, one player offers bets, another decides how to bet, and a third decides the outcome of the bet. We often call the first player Forecaster, the second Skeptic, and the third Reality.
How is this any different from the classical Dutch Book argument, that unless you maintain beliefs as probabilities you will inevitably lose money?
It’s just a different way of arriving at the same conclusions. The whole project is developing game-theoretic proofs for results in probability and finance.
The pitch is, rather than using a Dutch Book argument as a separate singular argument, they make those intuitions central as a mechanism of proof for all of probability (or at least the core of it, thus far).
You might be interested in some work by Glenn Shafer and Vladimir Vovk about replacing measure theory with a game-theoretic approach. They have a website here, and I wrote a lay review of their first book on the subject here.
I have also just now discovered that a new book is due out in May, which presumably captures the last 18 years or so of research on the subject.
This isn’t really a direct response to your post, except insofar as I feel broadly the same way about the Kolmogorov axioms as you do about interpreting their application to phenomena, and this is another way of getting at the same intuitions.
There’s a Q&A with one of the authors here which explains a little about the purpose of the approach, mainly talks about the new book.
I clicked this because it seemed interesting, but reading the Q&A:
How is this any different from the classical Dutch Book argument, that unless you maintain beliefs as probabilities you will inevitably lose money?
It’s just a different way of arriving at the same conclusions. The whole project is developing game-theoretic proofs for results in probability and finance.
The pitch is, rather than using a Dutch Book argument as a separate singular argument, they make those intuitions central as a mechanism of proof for all of probability (or at least the core of it, thus far).