Why I Believe Paul's Correction of Peter Defeats the Roman Catholic Claim to Papal Infallibility

The Catholic apologist may say that Galatians 2:11-13 and the incident in Acts 15 is not proof whatsoever that the Pope doesn't have the right or privilege to papal infallibility. They may say that Paul was simply giving brotherly correction. They may say that it would be just like a cardinal telling the Pope about being wrong on certain stuff. One situation they may want to compare is would be a cardinal tells the Pope something like, "Your holiness, you are wrong about my favorite food. I don't like pasta." This kind of problem happens because of a lack of understanding of what Roman Catholics believe about papal infalliblity. 

I agree with Roman Catholics in the sense that the Pope isn't perfect and that he can still make mistakes. The doctrine of papal infallibility doesn't teach that the Pope is always right. On the other hand, it means that the Pope is supposedly preserved from doctrinal error. Here's what the Vatican II Council says:
Vatican II explained the doctrine of infallibility as follows: "Although the individual bishops do not enjoy the prerogative of infallibility, they can nevertheless proclaim Christ’s doctrine infallibly. This is so, even when they are dispersed around the world, provided that while maintaining the bond of unity among themselves and with Peter's successor, and while teaching authentically on a matter of faith or morals, they concur in a single viewpoint as the one which must be held conclusively. This authority is even more clearly verified when, gathered together in an ecumenical council, they are teachers and judges of faith and morals for the universal Church. Their definitions must then be adhered to with the submission of faith" (Lumen Gentium 25).
Infallibility belongs in a special way to the pope as head of the bishops (Matt. 16:17–19; John 21:15–17). As Vatican II remarked, it is a charism the pope "enjoys in virtue of his office, when, as the supreme shepherd and teacher of all the faithful, who confirms his brethren in their faith (Luke 22:32), he proclaims by a definitive act some doctrine of faith or morals. Therefore his definitions, of themselves, and not from the consent of the Church, are justly held irreformable, for they are pronounced with the assistance of the Holy Spirit, an assistance promised to him in blessed Peter."
The infallibility of the pope is not a doctrine that suddenly appeared in Church teaching; rather, it is a doctrine which was implicit in the early Church. It is only our understanding of infallibility which has developed and been more clearly understood over time. In fact, the doctrine of infallibility is implicit in these Petrine texts: John 21:15–17 ("Feed my sheep . . . "), Luke 22:32 ("I have prayed for you that your faith may not fail"), and Matthew 16:18 ("You are Peter . . . ").

So why do I believe that Peter's error was a defeat on papal infallibility? Peter was not just wrong about something outside doctrine. It was indeed a doctrinal error especially with what appears to be him requiring circumcision as part of the Christian life for Gentile converts. If Peter shared his fallible opinion on non-doctrinal matters then the incident wouldn't have disproven papal infallibility. Instead, the incident further proved that papal infallibility is just another invention of the Roman Catholic institution to keep people in bondage. 

The whole idea is even ridiculous not because the Pope still commits human error but various Popes committed one form of doctrinal error after the other. The whole doctrine is nothing more than what Mark 7:8-13 condemns the very act of rejecting God's Word to keep their traditions. 

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