# Is there a procedure to derive models from axiomatic systems? [closed]

Is there a systematic procedure to construct a model of an 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.

2. We should not need to assume the axiomatic system is consistent 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.

• en.wikipedia.org/wiki/G%C3%B6del%27s_completeness_theorem – Matt F. Nov 6 '19 at 12:02
• A model can still exist even if the system is inconsistent --- you sure about that? Anyway, Google "model existence theorem". – Nik Weaver Nov 6 '19 at 12:03
• A structure with given properties may or may not exist, but there is no such thing as an inconsistent structure. Consistency is a property of axiomatic systems, not of structures. Also, the completeness theorem applies to arbitrary first-order theories, see en.wikipedia.org/wiki/… . – Emil Jeřábek 3.0 Nov 6 '19 at 13:50
• I have no idea what that question even means. Algorithms deal with finite strings (and other finite objects that can be encoded as finite strings). Models of interesting theories are usually infinite. – Emil Jeřábek 3.0 Nov 6 '19 at 14:06
• @IvanMeir. Models can't be inconsistent, that expression has no meaning. Also I think your understanding of "first order logic" is faulty --- e.g., ZFC is a first order theory. So yes, it is applicable to all of mathematics. – Nik Weaver Nov 6 '19 at 14:40