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.)


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