Interpretation vs model

Question

I was reading Introduction to the Foundations of Mathematics by Raymond L. Wilder and in it he says, and I paraphrase, that

an interpretation of an axiomatic system is the assignment of meanings or values to a given axiomatic system such that all statements of said system hold true. This results in the the statements of an axiomatic system to be true about a "concept". This "concept" is what we shall call a model of the axiomatic system.

What is really bothering me here is that I cannot grasp what a model is because the word concept seems too vague. Could someone please explain what a "model" of an axiomatic system is and how is it different from an "interpretation" of an axiomatic system?

Here is the link to the book (page 24 bottom section) :

https://books.google.co.in/books?id=Ma_zDZRHEa0C&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false

Answer 1

Different people's usage of the terminology here differs. But I think the following way of talking is pretty standard:

1. An interpretation is in the first instance an interpretion of a formal language $L$. Think of it as a map assigning truth-relevant values to the ingredients of $L$ -- for example, assigning objects to its names, extensions to its predicates, assigning a set of objects (a domain) for $L$'s quantifiers to run over, etc. (There is an assumed set of rules for determining the truth-values of sentences of $L$ in terms of such an assignment of truth-relevant values.) Derivatively, given a theory $T$ framed in language $L$, we can talk about an interpretation of that theory, meaning an interpretation of its language which assigns meanings-in-extension to the terms of the theory $T$.
2. We are then particularly interesting in those intepretations of the language of the formal theory $T$ which make $T$ axioms and hence (assuming its logical apparatus is sound!) its theorems all true. Such interpretations are said to be models of the theory. Thus understood, a model is a value-assigning map between a theory and some objects, extensions, sets, whatever. Though often we carelessly speak about the assigned objects, extensions, sets, whatever, as being the model.

In this way of talking, not all interpretations of $T$ are models of $T$, only those interpretations which make $T$ come out true about the assigned subject matter. But models are interpretations, particular 'good' instances of interpretations (or in the derived usage, are the objects and sets etc. mapped to by the good interpretations).

When Wilder says a model (in the standard sense I have just defined) makes 'the statements of an axiomatic system to be true about a "concept"' that's either careless or wrong. If I interpret a formalized theory to be about beer cans and strings connected them, then the theory is about beer cans and strings. If I interpret a formalized theory to be about numbers and addition and multiplication, then that theory is about natural numbers, addition and multiplication. Beer cans and strings aren't concepts: neither (on most views) are natural numbers and operations of addition and multiplication.

But we could, in the second case, sum up our intepretation of the formal theory by saying it is about arithmetic, and that we have modelled our theory in arithmetic, and even that arithmetic is our model. But it doesn't seem at all helpful to say that the "concept" of arithmetic is the model -- its the numbers and operation themselves which are the ingredients of the model.

Answer 2

An easy way to begin to understand the syntactic/semantic distinction closely related to the theory/model distinction is to ponder the way group theory works. You have the axioms developed already early in the 19th century, which is the syntantic side of the picture. Then you have a specific group satisfying those axioms, which is the semantic side.

When a given statement is interpreted in the model, it can be said to be true or false. For example, if $+$ denotes the group operation, then the statement that $1+1+1=0$ when interpreted with respect to a specific group will be either true or false.