I've already explicitly stated in my head post that I don't agree with formalism, mathematics is not limited to empty symbols rule following games, (usually called as string manipulation rules). Mathematics can involve meaning as far as those meanings are presented in a rule following manner, and this way it can provide analysis of those meaning, i.e. it can serve the application of concept analysis. But even then, the mathematics doesn't rely in those meanings, it lies in the consequential inferences those "meaning rule following games" derive, it is about how those meanings would be steamed by those rule following games. The nature of the concepts themselves is ought not to be a part of mathematics per se, it might be a part of philosophy of mathematics, but definitely it is not mathematics itself. Mathematics is just about consequential streaming scenarios, whether of meanings or of symbols.

to me axioms are just a medium in which consequential truths are harvested. Yet definitely in mathematical practice the axiomatics develop by capturing some concept, which fits the reciprocity part you've mentioned, and they are refined to suit basic intuitive grasps about those concepts at informal level, that way those axiomatic would be a kind of applied mathematics to those concepts, it provides analysis of those concepts. But all the systems born in that try (even the removed ones) are equally mathematical as far as they are media in which consequences are derived, that many times the original motivating concept is no longer the sole meaning attached to that system, and the line of development of those systems can ramify to capture different kinds of meaning quite different from the original meaning and so on.. Again the reality of those concepts are not the job of mathematics, it is the consequential load in those rule following games that is mathematics. I already alluded to that part of mathematical investigation in my last passage when I spoke about heuristics in finding the primary rules (axioms, inference rules, etc..) and suitable definitions and proof argument within those rule following games, and that part is, at many times, not directly trace-able to the primary rules themselves, but yet they are finally about consequences in those games, that part of mathematics is related to the ingenuity of mathematicians, but whatever it is, it finally aims to establishing an edifice in which consequential truths are harvested. They are creative acts that aims to enable consequential reasoning themes (games).

analytic truth truth by virtue of the meaning of the words of a statement, synthetic needs meaning and correspondence with reality as well. Here with this terminology I'm speaking about rule following which can even be of strings of empty symbols, so meaning is not involved here, however synthetic seems to be overlapping with receptive truths. I think that analytic is "meaningful consequential truths", so I think the term "consequential truth" is weaker than analytic truth, although of course you can object to this by holding that consequential truth is a kind of non-meaningful analytic truth or by saying that rule fellowship is a kind of meaning, you can call it meaning by having a role in following a rule, if so then we can subsume consequential into analytic. The new things is that KANT was saying that mathematics is apriori synthetic. Which this philosophy doesn't agree with. I more agree with Hume that mathematics is purely analytic nothing else. Regarding Quine's criticism he never criticized the substitutional form of analytic truths, on the contrary he considered it non problematic at all, he only criticized the meaningful kind of analytic truth. Here in this account the "consequential" part is more akin to the substitutional form of analytic truth although not necessarily limited to symbols, it can incorporate meaning but only if strictly manipulated by rule following games, which of course works by combinative-substitutional machinery. So there is some difference, as minor as it may be, but there is.

it depends on what kind of arithmetic you have in mind. Anyhow you can of course list a complete set of axioms for arithmetic that is not recursively enumerable, this is easy take for example PA+omega rule.

As regards provability and logical truth, one needs to be careful, what Godel has proved is that some theories cannot prove all statements made in their language, so there would be statements that the theory neither denies nor proves. That's all, this is about completeness of theories. From the perspective of this account this would be phrased as: some rule following games can have statements written in their language that has no consequential truth from the rules of those games nor do their negation has, because they are neither consequences of those rules nor is their negations. Godel demonstrated that for some theory T there can be a sentence S written in the language of T such that S is equivalent to statement "S is not provable in T", so obviously if T is consistent, then T cannot prove S. Be aware that this doesn't entail that S is a consequential truth of T, no! For S can be false and T be inconsistent! When we say that S is a logical truth, this is actually mean that S is a consequential truth of the theory T + Con(T), where "Con(T)" is the statement "T is consistent". In other words S is provable in T+Con(N), that's why its said to be a logical truth, in reality it means that it is a consequential truth from the rule following game T+Con(N), notice that it is not a consequential truth of T itself. "Logical truth" is provability in some system. So it is a relative concept. In other words S is not a logical truth of T, the logical truths of T are the theorems of T only.

I don't know what you mean that all propositions other than tautologies and contradictions has Correspondence truth. Correspondence with what? with Reality? what's Reality? I think this is mistaken, those propositions has no innate truths in them, rather truth is assigned to them by the system by dictation (if they are axioms) or by being consequential truths following from the rules of the game.

As regards your second reply, my answer is YES, inconsistent rule following games are definitely mathematical as far as our study of them is about the consequential truth of them.

I didn't get your question about how to create new rules, you simply stipulate them or they are consequences of newly stipulated rules? what's the problem?

I'll make a more elaborate response to this nice posting later. But for now I just want to resolve any misunderstanding that my prior response might have caused. What I'm really saying is that we can have many kinds of truths, there is a receptive kind of truth which is was you've mentioned by the correspondence theory, you can call it correspondence truth if you like, and there is another kind of truth which is what I call as consequential truth, you may consider it as a kind of coherence truth (or possibly the converse is true?! i.e. coherence truth is a kind of consequential truth, since formalism might have zero meaning but it does involve consequence, it's nice to explore that issue). The first kind is involved in empirical sciences, the second kind is involved in mathematics and logic (although I generally consider logic to be part of mathematics). The first answers to what the real world is? the second answers to what outcomes results from pre-specified rules. They are different concepts! I like the term "consequential truth" because I see it directly engaging the concept it negotiates. The term "provability" is more complex, it can even be said to be more or less a mathematical term, so I cannot define mathematics after it, since this would be a kind of a circular definition, but if we can manage cut from it the mathematical detail, then it might serve the same purpose of consequential truth. I also oppose to the idea of reductionism in mathematics, since this already proved to be false anyway. It is nice to reduce of course mathematical theories to simpler ones, but in my stance here all theories are equal, since all of them are just games, what's really important is the consequential truths in them. In nutshell you can say that:

Empiricism is about correspondence and Mathematics is about consequence.

I didn't get your point. Can you please elaborate further on it.

Thank you for your opinion about terminology about truth. But the essence of the matter of difference between us here (as it appears to me ) is just a matter of terming things. What you call TRUTH which is "correspondence with reality" according to the epistemology theory you've cited, I call as "receptive truth" or even it might be better called as "reality correspondence truth", I'm explicitly stating in my message that mathematics is not about that concept. I in opposition to your terminology prefer to use the term truth to denote another context which is quite different from the "correspondence with reality" context, and that context is what I've labeled as "consequential truth", you are free not to call it truth, you may term it as "consequentiality", or "consequential processing" which are fair enough. I explicitly gave an example where those two concepts collide. But to me I prefer to call it consequential truth, which is of course something that is quite different from correspondence truth. what I'm saying is that mathematics is about consequentiality. The mathematical systems are just mediums in which consequential processing is carried in. Also I traced back Analytic truths of I. Kant into consequential rule following processes. And I think mathematics is nothing but about such consequential machinery.

Although terminology is of course important for understanding, but I think one better try negotiate the essence of what's presented.