What is a theory and what is its extension As I understand, a theory is a set of sentences which are closed under some notion of deduction (i.e., applying deduction rules to the sentences of a theorem does not produce any new sentences) (wikipedia does not mention this notion of closure I think). 
In practice, a theory is represented by a subset of its sentences called axioms  such that all other sentences in the theory are deducible starting from these axioms and using deduction rules. These axioms are actually a representation of a theory, but not the theory itself. Sentences other than axioms in a theory are called theorems.
My Questions:

1) Is what I presented as a definition of theories, axioms, and theorems in the above correct? (I want to make sure that the notion of closure in the sense I defined above is necessary for the definition of theory or it is not?)
2) If we add an axiom to our existing set of axioms in a theory, does this new axiom extend the theory or it restricts it, or it depends on the axiom?

By extension of a theory $T$, I mean getting $T'$ such that $T \subset T'$ and by restriction I mean getting $T'$ such that $T' \subset T$
 A: 1) As we usually define them:

  
*
  
*A theory $T$ is a set of sentences such that $\varphi \in T \Leftrightarrow T \vdash \varphi$ (derivability closure).
  
*A set $\Gamma$ is the axiom set of a theory $T$ iff $T = \left\{ \varphi | \Gamma \vdash \varphi \right\}$
  
*If $T \vdash \varphi$, we say that $\varphi$ is a theorem of $T$ (that is, axioms are 1-line theorems)

2) Suppose you add $\alpha \notin T$ to the axiom set of $T$. The result is then a new theory $T'$, such that $\varphi \in T' \Leftrightarrow T\cup\{\alpha\} \vdash \varphi$. It is easy to see that $T'$ extends $T$, since $T \subset T\cup\{\alpha\} \subseteq T'$.
A: *

*This is basically all correct, except that I regard axioms as theorems (they're theorems with 0-line proofs). Ergo, the theorems of $T$ are just the elements of $T$. This is kind of like how if $V$ is a vector space, then the vectors of $V$ are just the elements of $V$.

*Adjoining a new axiom can never make the theory smaller, although it can potentially make it bigger.

*In my opinion, we should try to refrain from speaking of "axioms", since the term is basically meaningless. Admittedly, I tend to violate this recommendation all the time. Anyway, the way I see it, there are sentences. A collection of sentences can entail another sentence. If you have a collection $S$ of sentences, we can write $\mathrm{cl}(S)$ for the collection of all sentences entailed by the sentences of $S$. In words, $\mathrm{cl}(S)$ is the theory generated by $S$. That is all; in particular, calling the elements of $S$ "axioms" gains us nothing. For some reason, however, it seems very difficult to refrain from calling things axioms.
