Gödel's second theorem is something of an extension of Russell's paradox. Russell's paradox is a "hole" in his set theory. Gödel did not prove that Russell-Whitehead set theory (or, more accurately, Zermelo-Fraenkel set theory, which is the standardly-used formulation) is inconsistent. His work shows that it is incomplete.

The difference is that an inconsistent system can prove a contradiction: it can both prove a propostion and its negation! Such a system is most likely useless. An incomplete system is unable to prove some true proposition. This parallels the situation we all remember from math class: we know something is true, but are unable to prove it. However, unlike there, in this case our inability to prove a true statement is inherent in the set of axioms we use, and is not due to our necessarily limited talent.

Technical notes:

  1. Gödel's theorem actually applies to any "sufficiently advanced" system. However, even the full power of arithmetic is way more than is required for the theorem to apply; set theory is certainly sufficiently advanced.
  2. In fact, the theorem proves that if the system is consistent then it is incomplete. That is, there is some proposition P such that either both P and not-P are provable, or neither is!
  3. Worse yet, a "sufficiently advanced" system which can prove its own consistency is inconsistent...