Using logic to create theorems from a set of axioms

Question

I've recently become interested in the way that mathematical theorems can start from a very basic set of axioms and, through clever logic, be proven. I'm interested in knowing more about the reverse process. That is, given a set of axioms and the rules of (first order?) logic, derive theorems that are true, even if they are "uninteresting".

For concreteness, let's assume I have only the following two axioms for defining operations on a set of symbols:

There is a unique symbol, * in the set
There is an operation, @, defined so that, for any symbol o in the set $ *@o = o@* = p$ is a new symbol that is also in the set

What additional structure would I need in order to expand this (and similar) axiomatic systems into rich sets of theorems? Is it possible that an internally consistent axiomatic system can be chosen so that no theorems can be derived from it?

Not very clear... We have only one initial symbol : $$. Then we have a rule for producing what : strings of symbols or "equations" ? I mean, the result of the operation $@$ is e.g. the string $@*$ ? — Mauro ALLEGRANZA, Nov 06 '18 at 16:34
That's a valid point, and sort of answers the question - is it true that I need additional structure so that the mathematician is free to define symbols? For instance, is it "obvious" from the description that I could name my string @ something convenient, like Q, and use that new symbol to define yet further symbols? — Michael Stachowsky, Nov 06 '18 at 16:36
You can see the post Axiom Systems and Formal Systems for some ref. — Mauro ALLEGRANZA, Nov 06 '18 at 16:39
See also the post : Are axioms assumed to be true in a formal system ? — Mauro ALLEGRANZA, Nov 06 '18 at 16:40
Great, thanks! Out of curiosity, then, is what I describe above not a set of axioms but rather just a set of production rules, as in a formal language? — Michael Stachowsky, Nov 06 '18 at 16:41

Using logic to create theorems from a set of axioms

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