Riiiiiiiiiiight!
*pretends to understand*
Now, can you put it in layman's terms for me? (and by "me" I mean anyone who hasn't taken philo classes?)
Two of Goedel's more prominent works are his incompleteness theorems; the formal statement of the first is something like "any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete; for any consistent, effectively generated theory that proves certain basic arithmetic truths, there exists an arithmetic statement that is both true and not proveable in the theory". That boils down to "for any non-trivial formal system, there will be true statements that can't be proved in that system alone".
A nice example of this is in Cantor's diagonal argument, which he used to prove that there were varying cardinalities of infinity -- pretty much that there were multiple sizes of infinity. If you take
this table, you can read along the diagonal and invert the number (0 becomes 1, 1 becomes 0) to generate an element that isn't listed within the table itself. Extend that table to infinity in both directions and you can prove that the real numbers are, unlike number systems like the integers, rational numbers and so on, uncountably infinite. That's another, largely unrelated topic in itself, though.
The second is sorta similar: "for any formal, effectively generated theory including basic arithmetic truths, the theory contains a statement of its own consistency iff it is inconsistent". Long story short, "if a system's own logic allows it to be proved to be consistent, then there is contained within that system a contradictory theorem".