Difference between axiomatization and model

Question

As I study through set theory, I find the definition of axiomatization and models somewhat confusing.

The question is what is the difference between axiomatization and model?

Thanks.

Answer 1

An axiomatisation is a set of sentences—a theory—about some mathematical structure. For example, ZFC set theory is an axiomatisation of the cumulative hierarchy of sets.

A model is a mathematical object that satisfies—makes true—a theory. Usually it consists of a set, the domain, over which quantifiers in the theory range. Then there are relations and operations on that domain corresponding to the nonlogical symbols in the theory.

Answer 2

In brief, a model is a thing, while an axiomatization is a collection of sentences. For example, the natural numbers are a mathematical thing (a model), while number theory is a collection of statements (a theory).