Answer
The Roman Catholic Church teaches that the pope is infallible when he speaks from his position of authority on a particular issue or doctrine (speaking ex cathedra). Many misunderstand “papal infallibility” as indicating that everything the pope says is infallible. This is not what the Roman Catholic Church means by “papal infallibility.” According to the Roman Catholic Church, this infallibility of the pope, only when speaking ex cathedra, is part of the Roman Catholic Church’s Magisterium, or the “teaching authority of the Church” which God gave to the “mother Church” to guide her infallibly. This “teaching authority of the Church” is made up of the pope’s infallible teaching ability, the infallible teaching ability of church councils assembled under the authority of the pope, and the “ordinary” Magisterium of the bishops. This “ordinary” Magisterium involves, among other things, bishops in various places beginning to teach the same particular doctrine (for instance, the teaching that Mary was conceived without sin), and that if this teaching gains acceptance throughout the church as a whole, it is an indication that the Holy Spirit is working through the bishops and that this teaching is from God. The pope may later recognize this and proclaim infallibly that it comes from God and is to be accepted by all Roman Catholics.
The question is whether this teaching is in agreement with Scripture. The Roman Catholic Church sees the papacy and the infallible teaching authority of “mother Church” as being necessary to guide the Church, and uses that as logical reasoning for God’s provision of it. But in examining Scripture, we find the following:
1) While Peter was central in the early spread of the gospel (part of the meaning behind Matthew 16:18-19), the teaching of Scripture, taken in context, nowhere declares that he was in authority over the other apostles or over the entire church (see Acts 15:1-23; Galatians 2:1-14; 1 Peter 5:1-5). Nor is it ever taught that the bishop of Rome was to have primacy over the church. Rather, there is only one reference in Scripture of Peter writing from “Babylon,” a name sometimes applied to Rome, found in 1 Peter 5:13; primarily upon this and the historical rise of the influence of the bishop of Rome come the Roman Catholic Church teaching of the primacy of the bishop of Rome. However, Scripture shows that Peter’s authority was shared by the other apostles (Ephesians 2:19-20) and the “loosing and binding” authority attributed to him was likewise shared by the local churches, not just their church leaders (see Matthew 18:15-19; 1 Corinthians 5:1-13; 2 Corinthians 13:10; Titus 2:15; 3:10-11). Thus, the foundation of papal infallibility—the existence of the papacy itself—is not scriptural.
2) Nowhere does Scripture state that, in order to keep the church from error, the authority of the apostles was passed on to those they ordained (the Roman Catholic Church teaching of "apostolic succession"). Apostolic succession is “read into” those verses that the Roman Catholic Church uses to support this doctrine (2 Timothy 2:2; 4:2-5; Titus 1:5; 2:1; 2:15; 1 Timothy 5:19-22). Paul does NOT call on believers in various churches to receive Titus, Timothy, and other church leaders based on their authority as bishops, but rather based upon their being fellow laborers with him (1 Corinthians 16:10; 16:16; 2 Corinthians 8:23). What Scripture DOES teach is that false teachings would arise even from among accepted church leaders and that Christians were to compare the teachings of these later church leaders with Scripture, which alone is cited in the Bible as infallible. The Bible does not teach that the apostles were infallible, apart from what was written by them and incorporated into Scripture (2 Timothy 3:16; 2 Peter 1:18-21). Paul, in talking to the church leaders in the city of Ephesus, makes note of coming false teachers, and to fight against such error he does NOT commend them to “the apostles and those who would carry on their authority”; rather, he commends them to “God and to the word of His grace” (Acts 20:28-32).
3) Nowhere in Scripture is the “teaching Magisterium,” or mastery of bishops, taught and treated as of equal weight with Scripture. What history has shown is that, when any other source of authority is treated as being of equal weight with Scripture, that second authority always ends up superseding Scripture (such is the case with the Mormons' other accepted writings and the Jehovah’s Witnesses’ Watchtower). So it is with the Roman Catholic Church. Repeatedly, Catholic Catechisms state that many of their doctrines are not found in or based on Scripture (e.g., Mary being Co-redemptress and Co-mediator, sinless, and conceived without sin; Mary’s ascension; praying to saints and venerating them and images of them; etc.). For Roman Catholics, it is the “mother Church” that is the final authority, not Scripture, no matter that they say that the Magisterium is the “servant of Scripture.” Again, the Bible teaches that it is Scripture that is to be used as the measuring stick to determine truth from error. In Galatians 1:8-9, Paul states that it is not WHO teaches but WHAT is being taught that is to be used to determine truth from error. And while the Roman Catholic Church continues to pronounce a curse to hell upon those who would reject the authority of the pope, Scripture reserves that curse for those who would teach a different gospel from what had already been given and recorded in the New Testament (Galatians 1:8-9).
4) While the Roman Catholic Church sees apostolic succession and the infallible Magisterium of the church as logically necessary in order for God to unerringly guide the Church, Scripture states that God has provided for His church through:
(a) infallible Scripture (Acts 20:32; 2 Timothy 3:15-17; Matthew 5:18; John 10:35; Acts 17:10-12; Isaiah 8:20; 40:8; etc.),
(b) Christ’s unending high-priesthood in heaven (Hebrews 7:22-28),
(c) the provision of the Holy Spirit Who guided the apostles into truth after Christ’s death (John 16:12-14), Who gifts believers for the work of the ministry, including teaching (Romans 12:3-8; Ephesians 4:11-16), and Who uses the written Word as His chief tool (Hebrews 4:12; Ephesians 6:17).
In summary, the Bible speaks of only one abiding, "tangible," infallible guide left by God for His church. It is the written word of God, not an infallible leader (2 Timothy 3:15-17). And, as He gave the Holy Spirit to bear holy men along in the writing of those Scriptures (2 Peter 1:19-21), so He has given His Holy Spirit to indwell, fill, guide, and gift members of His church today for the purpose of directing His church through the proper interpretation of that written word (1 Corinthians 12 and 14; Ephesians 4:11-16). That there are schisms and false teachings today should be no surprise, for the Bible also warns us that there would be false teachers who would twist the written word (2 Peter 3:16) and that these false teachers would arise from within the churches (Acts 20:30). Therefore, the believers were to turn to God and the "word of His grace" for their guidance (Acts 20:32), determining the truth not by who said it, but by comparing it with the gospel already received by the early church, the gospel recorded for us in Scripture (Galatians 1:8-9; see also Acts 17:11).