Infallibility

Infallibility refers to the idea that something cannot be in error. In religious terms, it refers to a belief that the Catholic Church or the Pope cannot be wrong under certain conditions when teaching or instructing on the subjects of faith or morality. Based on the belief that the church hierarchy and the Pope have been entrusted to maintain the teachings of Jesus Christ, starting with Peter and continuing throughout history to any given current pope, the doctrine of infallibility supports and empowers an ecumenical council and/or a Pope to be final arbiters regarding any doctrine under review.

"Humanae Vitae [published 1968 and rejects contraception] was issued last week as far as the RCC is concerned and to roll it back, even though it wasn't declared infallible, would admit that the teaching of the magisterium is changeable. It will take a good 100 years of not talking about contraception at all before the RCC can quietly revise its view, and there is no way that it can get through that time without being asked about it. "

Infallibility effectively says that religious leaders must not be questioned.

Gödel's incompleteness theorems

Gödel's incompleteness theorems show that such infallibility cannot ever be established within any logical 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).