In order to prove the consistency of an axiomatic system we must come up with a model. Wikipedia gives the following definition for a model of an axiomatic system:
A model for an axiomatic system is a well-defined set, which assigns meaning for the undefined terms presented in the system, in a manner that is correct with the relations defined in the system.
Suppose that we have the following (simple) axiomatic system with "dog" and "green" as the undefined terms:
- There exists a green dog.
In order to check if the axiomatic system is consistent, a model is needed. I can't understand how the interpretation of the undefined terms is selected. Is it subjective? I mean there is no restriction on the interpretation of the term "dog". For example, we can assign two different meanings on the term "dog":
Concrete models
Case 1
"Dog" = common accepted meaning
"Green" = common accepted meaning
In this case the axiomatic system has no concrete model (there are no green dogs in the real world).
Case 2
"Dog" = frog
"Green" = common accepted meaning
In that case the axiomatic system has a concrete model (there is at least one green frog).
Based on the above examples can we say that an axiomatic system may have more than one concrete model just because different interpretations can satisfy the axioms?
Abstract models
What if we had to find an abstract model? Again we choose an interpretation for the undefined terms but this time according the above link:
A model is called concrete if the meanings assigned are objects and relations from the real world, as opposed to an abstract model which is based on other axiomatic systems.
I can't understand how we check if another axiomatic system satisfies the axioms of another axiomatic system (a model). Can someone give me a simple example of an abstract model?
Edit
In order to understand what a model is, I looked at this source. The book in chapter 1.5 (page 40 in the pdf) gives the following example of an axiomatic system:
- There are exactly three points.
- Two distinct points belong to one and only one line.
- Not all of the points belong to the same line.
- Two separate lines have at least one point in common.
Then it states that:
If we can find a model for a system—let’s call it system A—that is embedded in another axiomatic system B, and if we know that system B is consistent, then system A must itself be consistent. For, if there were two statements in A that were contradictory, then this would be a contradiction in system B, when interpreted in the language of B.
If we have found a model for an axiomatic system A which in turn means that A is consistent what is the point of looking if the axiomatic system B is consistent?
Finally, I looked at the following exercise to get an idea of what a model is.
Can we say that a model is whatever we imagine (apologize if this is not a technical term) that satisfies the axioms?
So, first of all, note that the phrase concrete model is marked as disputed on the Wikipedia article itself. The talk page itself however does not contain the word concrete, so I'm not sure exactly what kind of discussion around these terms is happening on Wikipedia.
The phrase concrete model is a little bit awkward, because a model constructed out of bits of math may or may not be concrete depending on one's perspective (see mathematical platonism on SEP for example). I'll use it though because alternative phrasings like real-world model are not much better.
A concrete model where objects are real-world entities and relations are real-world predicates is frequently used as a kind of pedagogical scaffolding to help people get familiar with the idea of a non-logical vocabulary that needs to be intepreted.
However, this scaffolding is often silently discarded pretty soon after it is introduced in logic books.
In the specific example you gave, where dog is interpreted as dogs-in-the-real-world in the first model and as frogs-in-the-real-world in the second, both the first and second structures are valid first-order structures.
Only the second, however, is a model of the theory there is a green dog. This is because the theory only holds in the second interpretation.