# difference between Axiom systems and a Model

In my understanding, system of axioms are a set of axioms such as PA, ZF, ZFC. For example, we can write when ZFC proves something :

$$ZFC \vdash \phi$$

However, sometimes, in logic theory, it introduces a Model T and writes like:

$$T \vdash \phi$$

Here, the terminology "Model" is same as an axiom system? If not, what is difference between a system of Axioms and a Model.

Usually, $$T$$ is notation for a theory, not a model. A theory pretty similar to a system of axioms, arguably the same thing. It is just a set of sentences, and $$T\vdash \phi$$ means there is a proof of $$\phi$$ using the sentences of $$T$$ as premises.
(The difference, if any, is 'system of axioms' has the connotation that the set of sentences is effectively describable in some way, whereas a theory is just an arbitrary set of sentences. Some authors also require that a theory be closed in the sense that if $$T\vdash \phi$$ then $$\phi\in T$$.)
Models are usually written as $$\mathcal M,$$ or something like that, and usually the notation is $$\mathcal M\models \phi$$ for a sentence $$\phi$$ or $$\mathcal M\models T$$ for a theory $$T.$$ Models are not at all the same thing as theories, rather they are mathematical structures in which the language can be interpreted. For instance we can interpret the formal sentence $$\forall x \exists y \exists z(x+y=z)$$ in the structure $$(\mathbb N, +)$$ (where $$+$$ is the addition operation on $$\mathbb N)$$ as "for every $$x\in\mathbb N$$ there are $$y\in\mathbb N$$ and $$z\in\mathbb N$$ such that $$x+y=z$$." The sentence is true in this interpretation.
The notation $$\mathcal M\models \phi$$ just means the sentence $$\phi$$ is true in the interpretation $$\mathcal M.$$ For theory $$T$$ (which recall is a set of sentences) $$\mathcal M\models T$$ means that all the sentences of $$T$$ are true in the interpretation $$\mathcal M.$$ In this case, we say "$$\mathcal M$$ is a model for the theory $$T$$."
One also sees the notation $$T\models \phi,$$ and this means that the sentence $$\phi$$ is true in every interpretation that is a model of the theory $$T.$$ Sometimes, when it's clear from context that we aren't talking about formal proofs, $$T\vdash \phi,$$ is also used to denote this. (In first order logic, the completeness theorem says $$\phi$$ is provable from $$T$$ if and only if it is true in all models of $$T,$$ so the two notions are equivalent anyway.)