Providing concrete models is more-or-less impossible, since we are not sure about many things in real world. On the other side, abstract models are usually insufficient for proving consistency, since the consistency of the other axiomatic system (which we have constructed the model in) needs to be proved first and the problem arises again!
Another method is to count all theorems of the system and make sure that they do not contradict.
So I have two questions:
- Can an axiomatic system be proposed whose theorems can be counted completely? Do you know an example?
- Is there another method to prove absolute consistency of an axiomatic system without providing a model? If no, why not? And if yes, may you please provide an example?