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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!

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    Do you believe that you are breathing air? In this Room? – 2017-01-31
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    @Socrates What do you mean? – 2017-01-31
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    A family $\Phi$ of sentences *axiomatizes* a class $\mathbb{K}$ of structures (say, the class of fields) if the models of $\Phi$ are exactly the structures in $\mathbb{K}$. That is, to show $\Phi$ axiomatizes $\mathbb{K}$, you need to show two things: - Every model of $\Phi$ is a structure in $\mathbb{K}$. - Every structure in $\mathbb{K}$ is a model of $\Phi$. – 2017-01-31
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    @NoahSchweber I don't even know! Maybe it would be possible to know something if we assume some things, and see how it plays out. I can't seem to find the keyword for this process. – 2017-01-31

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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.