In Godel's Incompleteness Theorem, his theorem is about a system of logic. Where can I find more about this study, especially the notation?

EDIT

I mean logic about systems in general. I worded the above a bit wrong.

-
It will not be much fun, the system is Principia, not current first-order logic. Also, he leaves out most of the detail. –  André Nicolas Dec 14 '12 at 21:50
@AndréNicolas That's a bit misleading. Gödel 1931 explicitly shows that his argument applies to any formal theory satisfying certain very general conditions (as we would put it, the system is primitive-recursively axiomatized and represents all primitive recursive functions). And the detail for this general version of the First Theorem is all there. –  Peter Smith Dec 14 '12 at 22:37
Do you want to learn about formal logical systems or Godel's theorem? –  Kaveh Feb 6 '13 at 21:52
Formal Logic Systems. Not any in particular, just in general. I do know that Gödel's theorems dealt with them in general. That is why I referenced his work. –  PyRulez Feb 7 '13 at 23:11