The first thing I'm getting wrong is Axiomatic Systems. Perhaps “models” or “theories” are better names?

Cauchy wrote:Logical systems are more self-contained than you're making them out to be, Treatid. A system comes equipped with:

1) Its alphabet, that is, the symbols that can be used to make its well-formed formulas (the logical sentences that could or could not be provable)

2) Its grammar, which is the collection of well-formed formulas, that is, the collection of sentences that the system could apply to

3) Its axioms, that is, the collection of well-formed formulas that are taken as proved

4) Its inference rules, that is, the mechanisms by which we determine which other statements are provable besides the axioms.

I think I'm understanding up to this point.

We need something to represent what we are talking about (symbols). A set of rules applying to the symbols (inference rules). A starting point (axioms). And the rules need to apply to the specific symbols we are interested in (grammar).

If we are missing any of these components – or the components don't match up (e.g. the rules don't apply to the symbols) then we aren't (cannot be) describing a single, definite, unambiguous system.

If we do have all these components then we are able to describe an axiomatic system.

It is the next bit where I'm getting lost.

It looks to me as though this divides everything up into one of two sets:

1. Well formed axiomatic descriptions.

2. Not unambiguous descriptions (sorry for the double negative).

And here I run into trouble. My thinking is as follows:

Given that axiomatic mathematics as a whole exists; it must belong in one of these two sets:

A. If axiomatic mathematics is not a well formed axiomatic system then we can't be sure what is being described.

B. If axiomatic mathematics is a well formed axiomatic system then it is subject to the Principle of Explosion.

However, since axiomatic mathematics is definitely a thing, and the principle of explosion doesn't apply to axiomatic mathematics as a whole there must be another option that I'm missing. But I can't see what it could be.

An obvious possibility is that there are other types of systems other than axiomatic description. Except that the basic assumptions of axiomatic mathematics (axioms?) as described by Cauchy above appear to me to preclude an alternative method of description.

a. If we have symbols, rules and a starting point then it is axiomatic mathematics.

b. If we lack symbols or rules or a starting point (or the rules don't match the symbols) then we have nothing.

What am I missing/misunderstanding?