Is there a systematic procedure to construct a [model](https://en.wikipedia.org/wiki/Model_theory) of an [axiomatic system](https://en.wikipedia.org/wiki/Axiomatic_system) from the system itself? 


For example given the abstract postulates of a ring we can show that the integers satisfies them and hence this is a model. Of course there are many other types of ring and clearly more general models. 


I want to know if there is any systematic algorithm or construction that could derived such models automatically, perhaps given knowledge of simpler mathematical objects like the integers, the reals, complex numbers for example.


Notes:


1. I'm definitely not expecting an algorithm that works in all circumstances as that is probably impossible like the [Halting Problem](https://en.wikipedia.org/wiki/Halting_problem).
2. We should not need to assume the axiomatic system is [consistent](https://en.wikipedia.org/wiki/Consistency) a model can still exist even if the system is inconsistent.
3. An algorithm that works for a restricted class of axiomatic systems would be interesting.



Is there a procedure to derive models from axiomatic systems?
A simple example for a restricted class is free algebras. Given a set of equations in a functional type (no relation symbols other than standard equality), one can construct the term algebra using the function symbols and variable symbols, and then construct congruence relations induced by identifying the left hand side with the right hand side of an equation. (Iterate this over the computable set of equations.) You always get a model, which may be the trivial one element model, from the quotient of the term algebra by the (equivalence relation determined from the) set of induced equivalence relations.


Gerhard "Is This A Good Example?" Paseman, 2019.11.06.


