so he essentially just reduced probability to measure theory--- so he did prove  the axioms but only in a way that he made it synonymous with some other quantity for which it is unclear as to what it means- not much better then a posit, a proof theory without a model theory, Hence why people still try to prove the axioms given certain interpretations (relative frequency), because its unclear that a measure as in a geometric sense really says anything about probability except as a synonym; or in some case by reducing it to logic----- that is what im trying to do- as an ordering over non probabilistic ranked indicative conditional- in the morgenbesser sense- where the principle of indifference is not assumed but proved in a strange way; in a way that forces you to be indifferent between equiprobable events because your belief will have the same truth value either way due to contextuality---- it uses counterfactual definiteness as is allowed by morgenbesser dominance conditionals even under indeterminism

I make the distinction in that quantum mechanics not only proves that the probabilities must follow a certain axiomatization, but even uniquely designates the values of probabilities--- even if there are multiple possible arrangments of values that satisfy addivity etc, it appears that logic alone, or some presupposition about the intepretation of probabilithy is built in, because it tells us what the values must be, and it proves the axioms and it does so by reducing it to another distinct quantity amplitudes which satifies the rules from which the probability rules are constructed. Its distinct I presume because these amplitudes must have some pre-probabilistic meaning if it would be any use to define probabilities in terms (ie in terms of the correlations they have in entanglements with other amplitudes, their relation ship to the eigenstate component and counterfactual results in different contexts- and the fact it is the square of the amplitude indicating that these entities obviously have meaning outside of the probabilities because two different amplitudes can have the same probabilities--- i gues it just encodes the possibly different entanglement correlations)

I very much like that quote from savage: I am also reminded by the quote of Russell to the effect that

' everyone agrees that mathematical probability is the most important and well used concept in science; but it is rather queer that it is so effective despite no one having the faintest notion as to what probability even means"

Well Van Fraassen and alan hajek (and others) argue that limiting relative frequencies, at least in certain circumstances are not borel fields, are not convex, compact, countably additive, or most importantly are not closed under disjunctions and conjunctionswhich apparently violates the requirement that probability functions are a field (ie the probability of A is defined and the probability of B, but not p(AB). This is why Van fraassen developed an idiosyncratic modal frequenstic account (1980 'modal frequentism' i think, which attempts to resolve these issues) . One can also see the issue of convexity arising in reichenbachs hypothetical frequentist interpertation in which every event in the horizontal sequence might have the same single case weight probability (vertical rel frequency=x) but this does not mean that the horizontal relative frequency=x. And this is an issue in general also for his metalinguistic partial entailment anologue of probability, called 'truth frequency' which is a logical probability relating the proportions of events A in the collective defined by the premises [B & P(A/B)=x]), ie each trial is an instance of [B & P(A/B)=x]), where P(A/B)=x is the physically necessitated or empirical frequency, and we count the number of such trials that are A's. The number of A's in this sequence of propositions divided by the number of propositions (distinct instances of B and P(A/B)=x) need not be x. Likewise, we can also see that Kolmogorov's strong law for independent but not identically distributed events may not even apply or actually may fail for frequency analysis if the infinite collective is instantiated by an infinite number of distinct reference classes each with only finitely many members, as the finitude of the number of members means they cannot make use of the limit frequency (the finite frequency might be anything)

In fact, Elliot Sober, i believe showed that under certain considerations this held for probability functions in general (probabilities of disjunctions not defined when disjuncts are), in effect showing that probability ab initio does not satisfy some of the more esoteric requirement of kolmogorov regardless of interpretation.

As for bayes rule, technically speaking unless you use the system of cox of de finetti, it not an axiom but a definition (that is why de finetti used conditional bets to justify it synchronically, whilst kolmogorov did not need to give any proof whatsoever). The propensity theorist can always use a system without bayes rule (popper renyi calculus where conditional probabilities are primitives, if they want to say that conditional probabilities are probabilities) or othewise could simply use the kolmogorov calculus as traditionally intended where conditional probabilities are not really probabilities are tall (but a piece of syntax that unfortunately uses the probability term).

As conditional probabilities are 'defined' to be ratio's of unconditional probabilities that could be construed as suggesting that this is all a conditional probability is; it is an entity with no pre- theoretical probabilistic substantive value of its own. Whereas were it an axioms, it would. Thus as traditionally defined, conditional probabitlies are just  a word that means the ratio of probabilties expressed in bayes rules. In that sense single case propensities satifsy bayes rule by taking the conditional probability as simply the ratio of the two propensities, but the fact that this conditional probability is not a propensity does not mean that propensities are not a satisfactory probabilistic interpretation (that is, by dint of not giving an propensity interpretation of conditional probabilties, and this is because there are no things as conditional probabilities as substantive stand alone probabilities).

This is because conditional probabilities are synonymous with bayes rule, and so they are not substantive probabilities on their own; conditional probability is simply a word which denotes that ratio (its just unfortunate that the word probability was employed in its defintiion), and this is what it means by something being a definition- its a piece of syntax, letters, not another independent probability entity analyzed or given truth condition in terms of bayes rule, it just is the ratio in bayes rule. It is not some stand alone entity which is a probability that has certain values given other probabilities, it just 'is' or 'means' that ratio of probabilities. So in some sense certain conditional probabilities are not really probabilities but synonyms for ratios, if one really analyze what is implied by kolmogorov calling the bayes rule a definition not an axiom.

A bigger problem is Milne's paradox which relates temporal and logical relations with the problems concerning probability on the single case (which concerns not explicitely propensity theories but any single case theory and so targets propensity theories and some other theories as well- only targeting propensity theories strongly in an indirect way insofar that they are tied more closely to probabilities being defined on the single case than in other interpretations- because single case propensity theory is really what one means if one is a full blooded propensity theorist, by which i mean it is truly that interpretation of propensity which distinguishes it from others, the long run form being arguably just an intensionally guided form of nomological frequentism; or a more mechanistic version of lewis humean frequecy account). This being opposed to the causal problem of humphrey paradxox that both forwards and backwards causal relations exist simultaneously for propensities which can be dissolved for the above reasons. MIller, McCurdy, Berkovitz, Aidan Lyon and Maurcio Saurez delve into the issue as to which problem is more pressing (the temporal single case issue which inflicts single case accounts or the humpreys causal issues which  inflicts propensities accounts, both long run propensities and single case propensities).

Finally the status of bayes rule is unclear in quantum mechanics to begin with; of course that does not mean the general issues still do not apply; approximate versions (of bayes rule) theirof, and rules such as luders axiom can play that role approximately in any case (and in any case, one still has milnes paradox). It not so much that causal relations must be forward in time, and conditional probabilities require backwards causation, its rather that conditional probabilities require that both an both forward and backward causation simultaneously, as per the time symmetry of probability and its unclear that such causal loops could be made sense (even if the probabilites can be made sense of, in quantum mechanics or otherwise)

OF courses its a number. Im saying its not a probability. Conditional probability just means the ratio, its just a word for that ratio, it isnt itself an actual probability

Checking the book (Google books gives pages 1-5 of Grundbegriffe der Wahrscheinlichkeitsrechnung), the model seems to be right there in §1 and §2 "Das Verhältnis zur Erfahrungwelt" and is distinctly classical: no problem having a set of events such that P(E)=1 (§1 axiom IV), and the conditions infinitely repeatable (§2.1). In §2.3 he gives the tossing of two coins as example, stating the 4 cases as the elementary events of E. Nothing about causation, so seems entirely a measure-based model to me.

He is also very clear that this elementary probability calculus is purely mathematical and distinct from reality. In paragraph 2 of the first chapter he writes (my own translation): "... this work is based entirely on these axioms and no considerations of reality of actual objects and events shall be made." ("Die Wahrscheinlichkeitstheorie als mathematische Disziplin soll und kann genau in demselben Sinne axiomatisiert werden wie die Geometrie oder die Algebra. Das bedeutet, daß, nachdem die Namen der zu untersuchenden Gegenstände und Beziehungen zu gehorchen haben, angegeben sind, die ganze weitere Darstellung sich ausschließlich auf diese Axiome gründen soll und keine Rücksicht auf die jeweilige konkrete Bedeutung dieser Gegenstände und Beziehungen nehmen darf") 

In the next paragraphs he states, repeatedly, that the theory and the elements of the set of events are entirely abstract.

If he used a more advanced model, it came later.

It seems to have been very clever of Kolmogorov to avoid trying to give an interpretation of "probability".

well he gave a model theory (which is supposed to be the sematnics and interpretation and truth conditions to some extent from which the axioms are proven) and thus gave an 'interpretation' if you can call it that but only in the trivial sense in that the measure is just in need of interpretation as probability itself, as it were (unlike say a frequency, where its pretty clear what that means)

Thus i call it the measure theoretical interpretation of geometrical one, which as close to a non-model/non interpretation as any interpretation/model can be. So i agree with you that he both did provide one (in that he gave a model) and yet in some sense he did not (as the model is not reducible to something antecedentally understood as most model are).

They say he was inspired by frequentism in particular von mises and reichenbach. BUt if you think about it really seems to me that he was inspired by the classical interpretation of probability, because thats essentially what the measure theoretical interpretation is: assuming every atom (point like continuous piece of space) in the probability volume has equal measure (to abuse every terminology- obviously why he wouldnt also have uncountable additivity). The probability volume being the box or square which represents the entire sample space with measure 1) where atom could be seen as an event with equal probability (lebesque measure) just as in the classical interpretation we have probability as a proportion of equally probable favourable events over all events; just that is more coarse grained.

It would also make sense in that the classical interpretation is probably the only one interpretation other than finite frequentism, which clearly satisfies the axioms perfectly (i say that tentatively as finite frequentism actually doesnt because it does not admit of irrational numbers-- so i guess the classical interpretation over a continnum would be the way to go, given he doesnt have the issue of additivity here as he has now uncountable additivty property).

The importance of the Kolmogorov axiomatic is that it opens the way to construction of the theory of random processes with such important applications as filtering, martingale theory, financial mathematics, stochastic control. Of course all these constructions might be derived from simple toss a coin model, if you like, but the beauty of good mathematical theory that it's not necessary.