TITLE: Interpretation vs model
QUESTION [6 upvotes]: 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

REPLY [2 votes]: 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.