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 equally-valued atoms, so I gave an example that is optimized towards their best case.
For small number of atoms my example can be trivially regraded by using straight addition on fraction parts after dividing fraction parts by the tripled sum of all fraction parts of all atoms.
I want also to say that axioms are not even compatible, because 0+0=0, and not strictly more than 0.
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
If we consider my addiditon on R, there is no escape.
There are a,b such that sum of any number of copies of a is smaller than b. No monotonous regrading can break this property. For normal addition, it is nonsense.
Well, their proof asks for lots of equally-valued atoms, so I gave an example that is optimized towards their best case.
I admit that I still haven't fully digested their purported proof. But does the proof require lots of equally-valued atoms? Or does it just accommodate lots of equally-valued atoms? (The regrading Θ doesn't have to be unique.)
I want also to say that axioms are not even compatible, because 0+0=0, and not strictly more than 0.
Yes, I think that they forgot to say that axiom 1 only applies when y is non-null. Axiom 1 is based on eq...
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.