a + b = [a] + [b] + arctg(tg(0.5 π {a}) + tg(0.5 π {b}))/(0.5 π)
This looks promising. At least, I don't yet see a general way to regrade it into normal addition. (I haven't digested Knuth and Skilling's purported general proof.) Can you make the counterexample more concrete? For example, what is a concrete set of images in R of atoms under the valuation such that this works as a counterexample? What is a minimal such set with respect to cardinality?
This is not isomorphic to addition because any sum of repetitions of 0.1 is smaller than 1.
It's not clear to me why this is a problem. You need to show that there is no regrading Θ satisfying Θ(a ⊕ b) = Θ(a) + Θ(b) for all a, b in the image of the valuation. (Here I use ⊕ to denote your addition above.) It's true that the ⊕-sum of any number of 0.1s is smaller than 1, but the image of the ⊕-sum under the regrading might be larger than 1.
I haven't digested Knuth and Skilling's purported general proof.
One of the many subtle problems with their proof is that they don't understand what is wrong with saying "n times a can be declared equal to na, we will regrade later". The problem is, of course, covering all the line.
Can you make the counterexample more concrete? For example, what is a concrete set of images in R of atoms under the valuation such that this works as a counterexample? What is a minimal such set with respect to cardinality?
Well, their proof asks for lots of equa...
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.