Q:

Hilbert systems and natural deduction systems in terms of "context"

I was reading the Wiki article on Hilbert systems and came across this passage:

A characteristic feature of the many variants of Hilbert systems is
  that the context is not changed in any of their rules of inference,
  while both natural deduction and sequent calculus contain some
  context-changing rules. 

What does this mean exactly?
Does a "context" in a Hilbert system refer to the set of assumptions/non-logical axioms we are building our proofs from in the first place?
For example if we have some proof starting off with $\Delta \vdash \ldots \,$ then we can basically use $\Delta \vdash$ in front of every line of our proof. Is this our "context" and is this what they mean by "it never changes"? And in ND we are able to change contexts?
What does this refer to exactly?

A:

The Wikipedia article refers to the fact that in natural deduction and sequent calculus there are inference rules that can change the set of assumptions (called context there) in a derivation. For instance, in both natural deduction and sequent calculus there is the following inference rule for introducing implication on the right of $\vdash$ (called $\to_\text{intro}$ in natural deduction, and $\to_\text{right}$ in sequent calculus):
\begin{align}
\dfrac{\Gamma, A \vdash B}{\Gamma \vdash A \to B}
\end{align}
Indeed, in the premise of such an inference rule the assumptions are $\Gamma, A$ (i.e. the formulas in $\Gamma \cup \{A\}$); while in the conclusion of such an inference rule the assumptions are just the formulas in $\Gamma$, the formula $A$ has been discharged from the assumptions.
It is true that in Hilbert systems the deduction theorem mimics the inference rule above (if $\Gamma, A \vdash B$ is derivable then $\Gamma \vdash A \to B$ is derivable), but it is a methatheorem about the system (it claims that if there is a derivation $\Gamma, A \vdash B$ in a Hilbert system, then there is another derivation $\Gamma \vdash A \to B$ in such a Hilbert system, with a completely different structure from the former), it is not an inference rule in the system, as is the case in  natural deduction and sequent calculus.

