How many times have you heard from well meaning, totally ignorant people that the Pope is just a man and can't be infallible because he is a sinner just like the rest of us? In other words, no man can be infallible because "all have sinned and fallen short of the glory of God". Being a sinner, which indeed the Pope is(he goes to confession just like the rest of us), has absolutely nothing to do with the biblical doctrine of infallibility. The doctrine of infallibility, officially defined at the Vatican I council of 1870, says that when the Pope is officially defining church dogma, the Holy Spirit is also. There are three requirements for infallibility to be invoked:
1. The pronouncement must be made by the official successor to Peter.
2. The subject matter must be in the area of faith and morals.
3. The Pope must be speaking ex cathedra (from the chair) of Peter, and must be intending to proclaim a doctine that binds the entire Church to assent.
If any one of the above 3 requirements is missing, the papal declaration is not considered to be an infallible doctrine. Not everything the Pope says is infallible.
So what are some of the biblical roots of the doctrine of infallibility? Well, for starters, Jesus Christ Himself, who created the Catholic Church in Matthew 16:18, also promised the gift of the Holy Spirit to guide it in truth always.
John 14: 16-17, 26: "I will ask the Father and he will give you another Paraclete—to be with you always; the Spirit of truth, whom the world cannot accept, since it neither sees him nor recognizes him because he remains with you and will be within you . . . . the Paraclete, the Holy Spirit whom the Father will send will remind you of all that I have told you"
John 16:14: "When the Spirit of truth comes He will guide you to all truth"
Luke 10:16: "He who hears you, hears me"
Mt. 16:19: I will give you the keys of the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven
Matthew 23:1-3: Then said Jesus to the crowds and to his disciples, "The scribes and the Pharisees sit on Moses' seat; so practice and observe whatever they tell you, but not what they do; for they preach, but do not practice.
(**NOTE - Just as the scribes and the Pharisees sit on Moses' seat, the Pope sits on the chair of Peter. And notice how Jesus told everyone to obey the scribes and the Pharisees, even though they were sinners. Just so, we have to obey the Pope in matters of faith and morals in the Church, even if the Pope is a sinner).
Then there was the Council of Jerusalem, in Acts 15. This Church council was held to determine whether or not Gentiles had to first be circumcised before becoming Christians. In the closing document, Peter says the following, from Acts 15:28:
"For it has seemed good to the Holy Spirit and to us to lay upon you no greater burden than these necessary things".
The Holy Spirit was invoked, so therefore, it has to be an infallible decision.
There have been 3 instances of an officially declared Papal Infallible doctrine. The first was in 1854, when Pope Pius IX declared the doctrine of the Immaculate Conception (The blessed Virgin Mary was conceived in St. Anne's womb free from original sin), then in 1870 at the first Vatican Council when the doctrine of Papal Infalliblity was officially declared to be true, and then in 1950 by Pope Pius XII when he declared the doctrine of the Assumption (the blessed Virgin Mary was assumed body and soul into Heaven).
So even if the Pope is a sinner, when it comes to officially defining dogmas concerning faith and morals, the Pope has the biblical protection of the Holy Spirit promised by Jesus Christ Himself to protect the Church from moral error. To ignore these scripture verses is to ignore one of the central tenets of the entire Bible. Even Protestants believe in the infallibility of scripture, because it's inspired by the Holy Spirit. This is no different. And some Protestants seem to make infallible statements all of the time about themselves being "saved", in spite of Phillipians 2:12.