The last thing I want to do is post a subtly wrong paper.
With a correct disclaimer (a nice construction, but skip the proofs as they are wrong) it could be still useful.
On closer look, Theorem from Appendix A is simply wrong. Trying to emulate polynomial multiplication inside real line pays off. I don't understand this. Can you elaborate a bit?
"a + b = [a] + [b] + arctg(tg(0.5 π {a}) + tg(0.5 π {b}))/(0.5 π)"
"[a]" is the floor of a, or the integer part of a, i.e. maximum integer no more than a. "{a}" is the fractional part of a, i.e. "a-[a]".
The addition of valuations between 0 and 1 is isomorphic to normal addition, but occupies only part of the real line; the addition of integer parts is simple and honest addition.
This is not isomorphic to addition because any sum of repetitions of 0.1 is smaller than 1.
This addition is not continious, but the paper claims that it doesn't need addition to be continious.
All-in-all, some of the "proven" formulas are actually pre-assumed in the paper. This was done on purpose. From the beginning, the author is trying to find nice axioms that will prove the things he wants to. I'm not sure this is a fair criticism (if I'm understanding you correctly).
Of course we wanted axioms to prove our favourite theorems. The problem is that these axioms are not enough to prove them, a proof with unstated additional assumptions is given and then the paper proceeds as if the theorems it needs were actually proven from the axioms.
I didn't mean epistemiological sense, I meant logical sense - once we picked the axioms, we should stop assuming our theorems are true and recheck them.
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 no...
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.