Infallibility is the idea that something cannot be wrong. In a Christian context, it refers to a belief that the Church or the Pope cannot be wrong when speaking under certain conditions. The conditions have been redefined occasionally - when it was proved that they were, in fact, wrong. However, when the wrongness may be in doubt - on moral issues, for example - it is usually said that the interpretation that was wrong, but the proclamation was true. Obviously, such an argument can given indefinitely.
Gödel's incompleteness theorems show that such infallibility cannot ever be established within any system. If a committee votes the pope to be infallible, they could be wrong in that declaration, as they are not infallible. Likewise, if they were, the reason they became infallible could be a fallible one. Kurt Gödel showed that systems cannot prove their own foundations unless those systems are contradictory (if contradictory you can prove anything).