The paper is somewhat inconsistence in its assumptions.
First, we say that we have only a finite number of atoms without anything known about them. Then in Appendix A we say that we can have an arbitrary amount of atoms with precisely-equal valuation. This is already suspicious.
On closer look, Theorem from Appendix A is simply wrong. Trying to emulate polynomial multiplication inside real line pays off. Theorem from Appendix B lacks some additional requirements (maybe monotonicity? at least?) and without these requirements it is false (and counterexample is of the same preversion weight class as the counterexamples in Appendix A that show that all axioms are needed). I have not chacked the Theorem in Appendix C because I am lazy.
All-in-all, some of the "proven" formulas are actually pre-assumed in the paper. While it can be useful to some people to build a highly-connected mental map of the concepts related to inference and probability (which is always good), it doesn't prove things from the axioms.
I did downvote the post from +9 to +8, although I appreciate the caution of the post's author in not calling anything in the paper true (only "claimed"), because, well, it is just a link to one more subtly-false to its claims paper.
First, we say that we have only a finite number of atoms without anything known about them. Then in Appendix A we say that we can have an arbitrary amount of atoms with precisely-equal valuation. This is already suspicious.
Please elaborate. I don't see anything suspicious in your paraphrase. For example, it makes sense to me that, if we don't know anything about the atoms, then we have the same knowledge about all of them, which corresponds to assigning equal valuation to all of them.
I've recently been getting into all of this wonderful Information Theory stuff and have come across a paper (thanks to John Salvatier) that was written by Kevin H. Knuth:
Foundations of Inference
The paper sets up some intuitive minimal axioms for quantifying power sets and then (seems to) use them to derive Bayesian probability theory, information gain, and Shannon Entropy. The paper also claims to use less assumptions than both Cox and Kolmogorov when choosing axioms. This seems like a significant foundation/unification. I'd like to hear whether others agree and what parts of the paper you think are the significant contributions.
If a 14 page paper is too long for you, I recommend skipping to the conclusion (starting at the bottom of page 12) where there is a nice picture representation of the axioms and a quick summary of what they imply.