We say a set of formulas G is an independent set if for all p in G, G-{p} (G minus p) does not entail p.
The question I'm having some trouble with:
Show that each finite set G has an independent subset D such that D entails p for each p in G.
What I don't understand is how you could define such an independent set, and what its independence has to do with every formula in G being derivable from it.
Any tips or pointers would be much appreciated!!
If $G$ is already independent, then you're done.
Otherwise there's a $p$ that is entailed by the rest of $G$. Remove that $p$ and proceed by induction on the number of formulas in $G$.
HINT
If G is not already an independent set, how can we successively take statements out of $G$ so that eventually the result is an independent set of statements yet still implies all statements of G as well?
Also: do you see why we have to take them out one by one?