Axiomatization in first order logic

Question

I am trying to understand what the term axiomatization exactly means . We didn't really discuss this in the lectures at my school but I have to write a homework related to that and I don't understand the question of the task. Can someone explain what the concept of axiomatization really means? My task is: Given a signature σ1 and a set of formulas Φ1 ⊆ FO[σ1] I have to show that Φ1 "axiomatizate" the class of fields . Of course I don't expect someone to solve the problem for me but I would really appreciate some easy examples of which steps should I follow to solve it (example of how to show the same thing not for fields but for something else) and what exactly this axiomatization concept means.

Thanks in advance!

Answer 1

An example is provided by the theory of groups, whose signature may be chosen to contain the binary function symbol $\circ$ (the group operation), the constant $e$ (the identity), and the binary equality relation. The theory of groups is then defined by the following axioms:

• $\forall x, y, z \,. x \circ (y \circ z) = (x \circ y) \circ z$ (associativity of $\circ$);
• $\forall x \,. x \circ e = e \circ x = x$ (existence of identity);
• $\forall x \,. \exists y \,. x \circ y = y \circ x = e$ (existence of inverses).

The choice of signature (and consequently axioms) is not unique. In the case of groups, one may include a unary function for the inverse, or omit the constant for the identity.